According to Steve Bates of the Sunday Mirror, the Reds are keen to build a squad capable of challenging for the Premier League title, with Virgil van Dijk arriving for a world-record fee and Naby Keita also set to come in in the summer.
"And that's why discussions are ongoing between [Liverpool owners] FSG and [manager Jurgen] Klopp to find a way to bid for Sanchez," the report continued. "Liverpool accept they are third in the queue and the biggest issue they face is matching the wages both Manchester clubs can pay Sanchez."
Bates added that while Liverpool will do what they can to try and convince the Chilean to move to Anfield, they accept it is a "big ask."
With his contract up at the end of the season, it's no surprise to see such a clamour to sign Sanchez, as he's been outstanding for the Gunners in recent seasons.
As these numbers from TifoFootball illustrate, the Chilean has made a massive impact since arriving in the Premier League:
According to Bates, City and United are said to be willing to pay Sanchez massive money to join their cause, with the latter ready to give him a weekly wage of around £300,000.
While it's a staggering sum, it's not a shock, as it isn't often a footballer of such pedigree is available for such a meagre transfer fee. Sanchez, after all, can play in numerous positions, can score and can create, too.
At this point, it appears the Reds are outsiders in the chase for the Arsenal man. But those who frequent Anfield will be encouraged to read Liverpool are reportedly going head-to-head with two of the division's biggest spenders in pursuit of one of the division's best footballers.
Reds Still in for Leon Goretzka
"Reports from Germany over the festive period suggested that a deal to take Goretzka to Bayern Munich at the end of the season was already done and dusted," his report noted. "However, contact has been established between the Merseysiders and player's representatives, and Anfield parties have been assured that all hope is not yet lost."
With Goretzka's contract set to expire at the end of the campaign, the Reds are allowed to discuss terms with him with a view to a possible switch in the summer on a free transfer.
Any side bringing him in for nothing would secure a major coup, as Goretzka is one of the most exciting young German players around. Journalist Callum Wright believes Liverpool may have some hope still:
Goretzka has been a force for Schalke in recent years, offering a dynamic and direct presence at the heart of the team's midfield. He can carry the ball through phases, find passes and score goals.
Bayern appear to be in the driving seat to secure the player at this point, and the attraction of a move to Munich is understandable for the 22-year-old. Yet while any deal remains unconfirmed, Reds supporters will still have slender hopes that the club can convince him to join up.