What job might Lord Hill get in the new European Commission?
David Cameron has been quite clear that he wants Britain to have a big economic portfolio.
According to the prime minister's spokesman, when Mr Cameron met the new Commission president Jean-Claude Juncker on Thursday, he "underlined his preference for securing an economic portfolio that would enable the British Commissioner to play a pivotal role tackling Europe's lack of competitiveness".
The problem is that it is not entirely clear what an "economic portfolio" means.
The list is traditionally understood to include trade, competition, internal market and services, and economic and monetary affairs.
But the latter always goes to a Eurozone country, some say energy should be included in the list too, and there is talk in Brussels that internal market and services could be divided into two separate jobs.
For now, the public strategy among British officials is to say that the UK is open-minded, that it would consider any of the economic jobs.
British diplomats are at pains to be accommodating to their European counterparts in an attempt to rebuild bridges after the battle against Juncker's presidency.
But I have been told that in private David Cameron is lobbying hard for the internal market job.
He wants Britain to have a role that will be central to his attempt to reform the EU.
Sources familiar with the discussions, however, have told me that Angela Merkel has made it very clear to the prime minister that Britain is not going to get internal market.
One source said that there are ten other countries lobbying for the job and Britain is low in the pecking order.
I understand that instead Chancellor Merkel is steering Mr Cameron towards making a bid for trade.
This is a portfolio that Britain had has before - Lords Mandelson and Brittan, Lady Ashton and Christopher Soames all held it - and it would be at the heart of the plans for an EU-US trade deal.
One source said there are also ten countries looking to get this job too. But despite this pressure, Mr Cameron is still holding out for the internal market job.
Having spent a day in Brussels at the latest summit, I have been struck by how many officials and diplomats say that Lord Hill's low profile and lack of experience will place him at a substantial disadvantage compared to other candidates, several of whom are former heads of government and senior ministers.
Some see it as a sign that Britain does not take Brussels seriously. Why could you not send a big hitter, they ask?
The one bit of luck that Lord Hill has had in the last 24 hours is that he has gained time.
European leaders failed to agree who should be the EU's next council president and high representative for foreign affairs.
Until those jobs are agreed, none of the other commission posts will be fixed.
There is not likely to be any decision until the next summit at the end of August.
That means that Lord Hill now has a long month to introduce himself to the people who matter in Europe's capitals.
If he can use that time to persuade Europe's leaders and officials that he is an homme serieux, a credible political operator with the prime minister's ear, and not a rabid Eurosceptic who will rush about issuing ultimatums, then he will do himself no end of good.
The other thing going in Lord Hill's favour is that many people in Brussels genuinely think Britain is teetering on the edge of withdrawal from the EU.
Whenever I go to Brussels, I am always struck how many there believe that the UK is in the departure lounge, in a way that most in Westminster do not.
But some Downing Street officials believe that they can use this fear to their advantage, allowing Juncker to believe that unless Lord Hill gets a good job, then the UK will take another step closer to the door.
Either way, there is a hard diplomatic slog ahead.