The 26-year-old England international will now discuss personal terms and undergo a medical at Anfield.
Liverpool manager Kenny Dalglish said: "We have been given permission to speak to the player.
"We will now try to agree personal terms with him, but it will all depend on the medical."
Liverpool had already had a £15m offer for the player rejected earlier this summer.
Villa chief Paul Faulkner said: "We've reached an agreement with Liverpool following an improved offer received on Wednesday and the submission of a transfer request by Downing.
"We believe there are replacements available in the market who can strengthen our squad."
Downing would become Liverpool's third British midfield signing of the summer, having already added Jordan Henderson and Charlie Adam to their squad.
Villa have already sold another of their highly-rated wingers in Ashley Young, who has moved to Manchester United.