Sky Sports News reporter Kaveh Solhekol, who has followed the Liverpool team on their pre-season tour of the USA, understands that Brendan Rodgers is trying to tie up a loan move for the Spain U20 star who is valued around £10m.
Arsenal have been in talks for the past fortnight with Manquillo over a similar season-long loan move, with a view to a permanent deal.
However, Liverpool look favourites to land the 20-year-old, who only featured three times for Atletico last season.
Sky Sports also understands that strikers Loic Remy and Divock Origi are due to have their medicals in the USA, with the signings to be announced by Liverpool on Thursday.
And, despite the denials of Southampton manager Ronald Koeman, it is believed Liverpool have tabled a £20m bid for centre-back Dejan Lovren.
At the club's Harvard base in Boston, Fabio Borini is training with the squad despite a £14m bid from Sunderland, as is Pepe Reina, who has returned from a loan spell with Napoli.
Defender Daniel Agger also trained by himself, but only due to a niggling injury.