Manchester City are reportedly considering a move for Barcelona captain Lionel Messi, arguably the greatest player in history.
The 33-year-old Argentina forward, who has won six Ballons d'Or and holds many goalscoring records, would clearly be the most impressive Premier League signing.
But which other of the recruits from abroad have had the 'wow' factor?
Take a look at some of the most exciting signings from the past and vote for your favourite. We also want your other suggestions so let us know via the comments section, which you can access when reading this story on the BBC Sport website, or tweet #bbcfootball.
Netherlands and AC Milan legend Ruud Gullit had a Ballon d'Or, a 1988 European Championship winners' medal, two European Cups and three Serie A titles when he joined Chelsea - long before Roman Abramovich's money - as a 32-year-old in 1995.
"Some people probably thought my legs had gone and I was coming here for the ride - to take the money and just get ready to retire. They were wrong," he said in a BBC Sport column earlier this year.
He was a success - especially after moving from sweeper into midfield - and took over as player-manager a year later, leading Chelsea to the FA Cup in his only full season in charge.
Juan Sebastian Veron
Argentine Juan Sebastian Veron was one of the most famous playmakers in the world when Manchester United paid £28m to sign him from Lazio in 2001. BBC pundit Mark Lawrenson said the signing meant United were "unstoppable".
Despite an encouraging start - winning the September Premier League player of the month award - he struggled to adjust to English football and two years later joined Chelsea. He had one injury-hit season there and never played in England again.
Manchester United signing a 34-year-old free agent would usually not be seen as a great signing. But former Barcelona, Juventus and Paris St-Germain striker Zlatan Ibrahimovic had scored 392 goals and won a trophy for 15 consecutive years when he joined United in 2016.
He scored 28 goals in 46 games in a successful season, winning two cups, but rarely played the following year after a serious knee injury, before joining LA Galaxy.
Manchester City beat Chelsea to the signing of Brazil forward Robinho, then 24, from Real Madrid for a British record £32.5m, just hours after their Abu Dhabi takeover was completed in 2008.
Chief executive Garry Cook told BBC Sport in 2018 they had 24 hours to find and buy a marquee signing on deadline day or else the takeover might not happen.
It did not start well as Robinho accidentally(?) said he had joined Chelsea in his unveiling news conference. Just 16 goals and two years later he had gone to Milan for £15m.
German World Cup winner Jurgen Klinsmann arrived to a mixed reception when Tottenham signed the 30-year-old striker from Monaco.
A self-deprecating dive celebration on his debut helped him quickly become a fans' favourite and he finished the season with 21 Premier League goals. He left for Bayern Munich after one season but came back in 1998 and ended his top-level career with Spurs.
Carlos Tevez and Javier Mascherano
West Ham signing Argentina forward Carlos Tevez and midfielder Javier Mascherano, both 22, from Corinthians via third-party owners remains two of the most surprising Premier League transfers - with many top European teams having been interested in the pair.
Mascherano failed to beat Hayden Mullins to a regular starting spot, while Tevez scored seven goals - all in the final three months of the season - including a goal on the final day against Manchester United which kept them up, despite legal complaints from Sheffield United.
Both players went on to have hugely successful careers. Tevez won Premier League titles with Manchester United and Manchester City, while Mascherano moved on to Liverpool and then Barcelona.
Italy striker Fabrizio Ravanelli, then 27, became the third most expensive Premier League signing when he joined Middlesbrough for £7m, just two months after scoring in the 1996 Champions League final for Juventus.
He introduced the shirt over the head celebration to England and scored 31 goals - including a hat-trick on his debut against Liverpool. His season at Boro was eventful - they lost the FA Cup final, the League Cup final and got relegated. He left for Marseille after a year at the club.
World Cup-winning France defender Marcel Desailly, 30, had won the Champions League with two different teams when Chelsea signed him from AC Milan for £4.6m in 1998.
He played 216 times over six years for Chelsea - some in midfield - although the only major trophy he won was the FA Cup.
Netherlands midfielder Edgar Davids, 32, was past his best when he joined Tottenham on a free transfer after a successful career with Ajax, AC Milan, Juventus, Barcelona and Inter Milan.
He played 44 times - scoring once - over a year and a half - and his time at White Hart Lane is probably best remembered for getting punched by Robbie Keane in training.
Verdict: Neither hit nor miss
There was a lot of hype when Juventus and France midfielder Paul Pogba, 23, joined Manchester United for a world record £89m in 2016 (well, rejoined as he was at United as a youngster).
He has scored 32 goals in 157 appearances but has suffered a few injuries and has only won the Europa League and EFL Cup in four years at Old Trafford.
Verdict: Neither hit nor miss
AC Milan and Ukraine striker Andriy Shevchenko, 29, was one of the world's most respected goalscorers - and 2004 Ballon d'Or winner - when Chelsea signed him for £30m in 2006.
He only managed nine goals in 47 Premier League appearances - winning one League Cup and losing a Champions League final - before returning to Milan two years later.
Angel di Maria
Manchester United paid a British record £59.7m to sign Real Madrid and Argentina midfielder Angel di Maria, 26, in 2014.
He started well but after only four goals in 32 appearances left for Paris St-Gemain a year later, with United taking a £15m hit on the fee.
Ballon d'Or winner George Weah was 33 when he joined Chelsea on a six-month loan from AC Milan in 2000.
The Liberia striker scored five goals in 15 games, including a late debut winner against Tottenham, and won the FA Cup. He went on to have a less successful spell at Manchester City and is now president of his country.
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