Press Association Sport understands Downing, who has had few first-team opportunities under Brendan Rodgers, would not be averse to a loan move back to the Riverside Stadium as he looks for regular football.
Liverpool seem likely to attempt to offload the 28-year-old, who joined the Reds in a £20million switch from Aston Villa in July last year, in January with Rodgers keen to address his lack of strike-power.
There is interest from a series of Premier League rivals in a permanent move but Downing is understood to still be determined to prove he has a long-term future on Merseyside.
In the circumstances, the player would prefer a temporary move, and his former employers would provide a tempting alternative.
Boro, who currently lie in third place in the Championship, were not commenting publicly on the matter, but sources on Teesside have indicated that nothing is likely to happen ahead of Thursday's emergency loan deadline.