The two clubs have reached an agreement for the winger, who has been a prime target for Dalglish all summer, after two previous bids of £12 million (Dh71m) and £15m had been refused prior to yesterday's breakthrough.
The Liverpool squad arrived in Malaysia today in their on-going pre-season Asian tour but an official representing the Malaysian sponsors, speaking on condition of anonymity, said they have been told Dalglish will fly back to England later today.
Downing effectively forced Villa's hand by submitting a formal transfer request and the Birmingham club have now accepted a bid, which has not been disclosed by either club but was last night reported to be worth £20m, for the England international.
Negotiations can now begin with the player, who turns 27 later this month.
"Liverpool have now been granted permission to discuss personal terms with the player and arrange a medical," Liverpool said in a statement on their website.
Alex McLeish, the Villa manager, had insisted he wanted Downing to stay in the midlands, as Liverpool hovered with intent. But the size of the latest offer proved irresistible, given Downing has only two years remaining on his contract.
Downing made it clear only last month, when on England duty, that he would not sign an extension with Villa, effectively forcing them to listen to offers.
Arsenal were reported over the weekend to have joined the competition for left-footed Downing, who can operate on either wing.
However, Villa denied receiving an offer from the Gunners, and barring a late intervention from Arsene Wenger the path appears clear for Downing to move to Anfield.
Should the deal be concluded, the international, capped 27 times by England, would become the third high-profile midfield recruit by Dalglish this summer, following the signings of Jordan Henderson from Sunderland and Charlie Adam from Blackpool.
News of Downing's anticipated arrival came on the day Liverpool offloaded Paul Konchesky, a summer signing 12 months ago, to Leicester.
With Milan Jovanovic also expected to leave, perhaps for Anderlecht, and others including Christian Poulsen, Alberto Aquilani and David Ngog thought to be available for transfer at the right price, Dalglish is rapidly rebuilding the squad he inherited from Roy Hodgson in January.
Dalglish's team won 4-3 in a pre-season tour match in China against Guangdong yesterday, in which both Poulsen and Ngog scored.
Downing's departure from Villa after just two years would be a setback for the midlands club, who also sold Ashley Young to Manchester United last month.
McLeish will now press ahead with plans to bring in new blood.
The Scot has already been linked with Wigan Athletic's Charles N'Zogbia, whom he came close to signing when in charge of Birmingham, while the goalkeeper Shay Given's £3.5m move from Manchester City is expected to go through soon.
"We believe there are replacements available in the market who can strengthen our squad ahead of the coming season, and we are in discussions as such," said the Villa chief executive Paul Faulkner.
"We are also working hard to conclude a deal for a goalkeeper who we believe can also strengthen the side."