"I think Paul's OK for the time being," chairman Corney told BBC Breakfast on Monday after his club booked their place in the fifth round for the first time in 19 years.
"Paul and I have a great working relationship and Paul puts pressure on himself. Hopefully this will be the kick-start and we won't be talking about Paul's position again."
Asked on BBC Radio 5 live if Dickov would still be in charge for the Everton FA Cup fifth-round tie on February 16, Corney said: "I would love it but we have got one point out of 24 and there are probably three or four games before the Everton match.
"I desperately want Paul to be in charge. He's a friend, a great manager and he's got a really bright future in the game. I actually think it might be hard for us to hold on to him."
Speaking on BBC Radio Manchester, Corney added: "I have to use my head as well as my heart. Days like Sunday are fantastic but you can get carried away and you have to make the right decisions for the club.
"If you'd have asked me at the start of the season where this team could finish, I would have said certainly in the top half and possibly in the top six. To watch us struggle at the bottom is heartbreaking for me."
Dickov said on Sunday that he planned to speak to his chairman about the future. "There are a lot of things that went on that me and Simon need to talk about," he said.
"You can spit your dummy out or you can roll your sleeves up and get on with it and one thing I will always do is work hard."
Two goals from Matt Smith and a header by Reece Wabara secured a famous win which Dickov dedicated to the club's fans.
"There are a lot of emotions at the minute," the Latics boss told BBC Radio 5 live.
"I am very proud and I am very pleased for the fans as it has been a long time without success for them.
"We have lost seven out of eight but they have been by the odd goal.
"We have not been getting any luck, but the performances have been OK, and we got the bit of luck we needed.
"I was a lucky boy to have the playing career I had, but this is the top moment."
It was the first time Liverpool had been beaten away from home in the FA Cup by a team at least two divisions below them since defeat at non-league Worcester City in 1959.
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