Liverpool will follow up the signing of Charlie Adam with an improved offer for Stewart Downing - but they still face a fight to persuade Aston Villa manager Alex McLeish to sell the England winger.
Villa turned down Liverpool's opening £15m bid and BBC Sport understands they are refusing even to consider a price for Downing, despite speculation that they will sell for £20m.
Downing is expected to hold further talks about his future with McLeish on Friday and Liverpool's improved offer, likely to be around £18m, could even have arrived by then.
Liverpool boss Kenny Dalglish is keen to make the 26-year-old his third signing of the summer after concluding a deal for Sunderland's Jordan Henderson and agreeing a fee for Blackpool's Adam, and the player is eager to make the move to Anfield, particularly since talks on an extended contract at Villa Park stalled.
Dalglish leads Liverpool on an eight-day Far East tour on Sunday and would hope to have a deal wrapped up by then but Villa are in no mood to part easily with Downing, after already losing Ashley Young to Manchester United for £16m.
Villa are prepared to take a hardline policy on Liverpool's interest in Downing, who has two years left on his contract, and insist they do not want to sell, dismissing suggestions that there is a price that would persuade them to change their stance.
McLeish said: "He had a magnificent season last year and really came to full maturity as a top player. I understand why there are teams interested in him and that is what happens when you play well consistently.
"His performances last season were top drawer. That's why he's not for sale. The club don't want to sell him. I spoke to him briefly last week and I intend speaking to him again when all the lads are back in for training."
Downing, who joined Villa from Middlesbrough for £12m in July 2009, may well have to force the issue by putting in a transfer request if McLeish continues to block any further moves by Liverpool.