Jurgen Klopp is looking to bolster his forward line with the future of Anfield misfit Daniel Sturridge in doubt and has turned his attention to the Mexican.
Jimenez has scored eight times in 18 games across all competitions for Benfica so far this season and Liverpool understand that while he is going nowhere in January, he could be available at the end of the season.
The 25-year-old won the Spanish Supercup with Atletico Madrid in 2014 and helped Benfica win a league and cup double last season.
He was expected to join West Ham when he left Atletico in August 2015 but failed to show up for his medical claiming he had missed his flight to London after oversleeping.
Benfica then signed him in a two-stage deal which eventually became worth an estimated £18m when they finally bought out the remaining 50 per cent of his rights from a third party last summer.
That represented a club record and the Portuguese giants, who are already making contingency plans should he leave, are looking for a big return on their investment.
Jimenez has impressed in England before. He won a gold medal with Mexico at the London 2012 Olympics, playing against Brazil in the final. He has nine goals in 48 appearances for the senior team.
Liverpool see him as an ideal candidate to freshen up their squad and believe he has the potential to become a star at Anfield.
Jimenez can play as a centre forward but is also comfortable in a wider role and his versatility appeals to Klopp, who likes his forward players to constantly inter-change positions.
Jimenez helped Benfica reach the Champions League quarter finals last season, scoring in their eventual defeat to Bayern Munich.