Inter Milan have also been chasing a deal for Keita, who ESPN claim is seen in Italy as a potential replacement for their coveted Croatian international.
Radio Sei claimed two weeks ago that the player had left Lazio's pre-season training camp in Auronzo di Cadore early amid growing rumours that he was set to negotiate terms over a switch to Inter.
Such a deal has yet to be completed, however, with long-term suitors Juventus now stepping up their attempts to acquire Keita.
Italian newspaper Leggo suggested on Tuesday (25 July) that his valuation had increased from £22m ($28.7m) to around £27m following an approach from the reigning Serie A champions that has now been confirmed by CEO Giuseppe Marotta.
"We have made an offer to [Lazio President Claudio] Lotito for Keita and he will now evaluate it," he told Mediaset Premium. "We'll see what happens."
Inter's last bid for Keita, who has also been linked with the likes of Premier League trio Liverpool, West Ham United and Everton as well as AS Monaco, was said to be worth €22m plus a further €3m in add-ons. However, that was apparently rejected as being too low.
It was previously claimed that the Spanish-born Senegalese had a preference for Turin, but that Lotito was only willing to negotiate with Inter. Keita's agent, Roberto Calenda, revealed in early June that Lazio had reached an agreement with big-spending AC Milan over the transfer of a player who is out of contract next summer and has yet to agree an extension.
If Juventus did succeed in signing Keita, then it would presumably provide another sizable setback to United's already complicated attempts to acquire Perisic. Mourinho has been open about his desire to add another wide player to his squad before the end of the transfer window and hoped to complete a deal for the 28-year-old before Inter flew to China last week for their pre-season tour.
However, United's refusal to meet a £49m asking price meant he travelled to the Far East with the rest of the squad. Further progress has been slow, although new Inter boss Luciano Spalletti did admit earlier this week that the club would consider "difficult to reject" offers for Perisic. However, he also stressed the need to source a "valid alternative".
Addressing the situation again in a post-match press conference that followed Inter's 2-0 win over Bayern Munich in Singapore on Thursday, a match in which Perisic played 84 minutes and notched an assist, Spalletti said: "For me the talk is always the same when it comes to him. He's strong, he's a player I want to keep with me, but then we must consider the opinion of the club and the player."