Liverpool are understood to have made a £15 million bid for Joe Allen, triggering a release clause in the Swansea midfielder's contract.
Reds boss Brendan Rodgers is hopeful that the offer - an increase on an earlier £12.5 million bid - will land the Wales international, who represented Team GB at the Olympics.
Allen, 22, has been a Liverpool target since Rodgers left Swansea for Anfield in June.
The Northern Irish manager had agreed not to make any approaches for Swansea players for 12 months as a condition of his move to Liverpool. However, a technicality in Allen's contract allows him to join one of five clubs, including Liverpool, for £15 million, and with the Merseysiders having met the asking price he is free to hold talks over a move.
Rodgers has made no secret of the fact that he is keen to add to his squad, with his only signing so far this summer being striker Fabio Borini, who arrived from Roma in an £11 million deal.
"I'm very reluctant to speak about players at other clubs," Rodgers said. "I know I've got a bit of work to do. We need to improve the squad - there's no question about that.
"But, unfortunately, it takes a wee bit of time. It's not as easy as maybe some people think, especially when you're a club of this status. So it's a little bit more complex - but certainly my intention is to bring in a few more players.
"I'd love them to be here sooner rather than later, but the most important thing is that they're here by the end of August, and then we know the group we're working with until January.
"We've got some very good players here, and I want to give them a chance. But this is a club people want to be at, and I'm hoping a number of those will come in over the next couple of weeks."
Earlier this week, Swansea manager Michael Laudrup had warned Rodgers that Liverpool would have to fork out £20 million in order to take Allen to Anfield.
"Joe is worth a lot of money," Laudrup said. "Their manager knows if he wants to change the way Liverpool play, he needs a midfielder like Joe who is young and can play at the highest level.
"We saw they bought a couple of midfielders in the last few years for a lot of money but Joe is on a level above. If one is worth £17 million then he is worth more, perhaps £20 million or so."