The 17-year-old attacking midfielder has emerged as one of the most sought-after young players in Mexico after establishing himself in Tijuana's senior team.
Mora has scored five goals and provided one assist in seven league appearances this season, while his progress has also earned him international recognition with Mexico.
Liverpool are among a long list of major European clubs linked with the teenager, including Arsenal, Manchester United, Bayern Munich, Paris Saint-Germain, Barcelona and Real Madrid.
Reports have previously suggested Tijuana could demand as much as £40 million for Mora, while the player is understood to favour a move to one of Spain's two biggest clubs over a switch to the Premier League.
However, local journalist David Faitelson has now claimed Liverpool have made a significant move.
“I have been told in Tijuana that they have received, through Gilberto Mora's representative Rafaela Pimenta, an offer from an English club – in this case Liverpool,” Faitelson said on TUDN.
“It would be a historic sum – more than $50 million – for Tijuana for Gilberto Mora, who turns 18 next week.”
The report has not been confirmed by Liverpool or Tijuana, but the claimed figure would make any transfer a record sale for the Mexican club.
ESPN Mexico has also reported that there is concrete interest from several clubs and that one European side has progressed further than the rest, although it did not identify the club.
According to the report, negotiations involve Tijuana owner Jorgealberto Hank, Mora's representatives, the interested European club, the player and his father.
Mora has already made 62 senior appearances for Tijuana, scoring 15 goals and providing three assists. He has also scored once in 13 appearances for Mexico.
His contract with Tijuana runs until the summer of 2029, giving the club a strong negotiating position if European interest develops into a formal transfer.
Liverpool's reported bid would represent a major statement of intent, but Mora's future remains unresolved while several of Europe's biggest clubs continue to monitor his development.