Barry was also linked with a move to Arsenal on Tuesday
Unsettled Aston Villa captain Gareth Barry will return to training on Wednesday, according to the club.
Barry had previously been ordered not to report for pre-season training after criticising manager Martin O'Neill in an unauthorised newspaper interview.
However, an unfit Barry is unlikely to play against Danish side Odense in the Intertoto Cup on Saturday.
The midfielder, 27, has been frustrated by O'Neill's decision to reject a succession of bids from Liverpool.
The Premiership rivals have been unable to agree terms for the proposed move, with the Anfield club still to meet O'Neill's demands.
Villa, who also fined Barry two weeks' wages after his outburst, are understood to want £17m plus full-back Steve Finnan, as well as full payment in the next 12 months.
"Liverpool have come back to us and we have responded to their proposal with a letter. We're still waiting to hear back," O'Neill said.
"It's really up to Liverpool. I really don't know how it's going to work out at the moment. I'm sure there will be some sort of resolution at some stage or another."
The latest sticking point could be the value of Republic of Ireland international Finnan, who Liverpool rate at £2m. Villa believe he is only worth half of that figure as he enters the final year of his contract.
Villa are also believed to want Liverpool to follow the traditional Premier League structure of paying half of the fee now and the rest in 12 months' time.
O'Neill wasted no time in rejecting Liverpool's last offer, which was in the region of £15m.
O'Neill told the club website: "We put a valuation on him and Liverpool value him differently at the moment. That's what the stalemate is, nothing else.
"When somebody says they want to leave there's not much you can do. If they meet our valuation he will go.
"Gareth has pointed out he wants to go and has made his position clear, that has been the case since the beginning of June, when he said he wanted to join Liverpool."
It has been suggested that Liverpool's interest in Barry has waned, what with the mooted sale of Xabi Alonso to Juventus falling through. Reds boss Rafael Benitez has also admitted an interest in Tottenham striker Robbie Keane.
I'm desperate to play Champions League football and that's why I have to leave Villa
In the interview that led to his punishment, Barry admonished Villa for not trying hard enough to keep him - even though O'Neill insisted they want him to stay.
The player signalled his intention to move when he complained about Villa's lack of urgency in speaking to him.
"There's no going back, it's time for me to move on," he said.
"I'm desperate to play Champions League football and that's why I have to leave Villa."
But O'Neill vehemently denied Barry's claim, saying: "The chairman and I were surprised at Gareth's comments," he said.
"Contrary to what he said, we desperately wanted to keep him - and he knows that.
"Why on earth would we not want to keep our very top player at the football club when we're trying to improve?"
Barry captained England for the first time and scored his first international goal during the recent friendly against Trinidad and Tobago.
Liverpool are keen to partner Barry in midfield with England colleague Steven Gerrard.
Meanwhile, defender Martin Laursen is in the frame to take Barry's job as Villa captain.
O'Neill said: "There are a number of people on the list and Martin Laursen would certainly be one of the names I'd be thinking about.
"He was terrific last season. There are a number of people who would want it but there are a number of people who are on my list who might not want it. I haven't discussed it yet.
"There are quite a few candidates at the club and perhaps others not yet at the club so that's something I'll look at further down the line."
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