Miguel Herrera has been fired as Mexico coach following allegations that he punched a television reporter.
Former Arsenal midfielder Abou Diaby has rejected an offer to join West Brom in favour of finding a new club in France.
The 25-year-old is expected to feature heavily for England at the World Cup finals in Brazil after a superb season for the Saints.
Lallana, recently voted Southampton's Player of the Year, has scored nine goals in 37 Premier League games this season.
He also has six assists and has created 69 chances for his team while earning regular England call-ups alongside team-mates Rickie Lambert, Jay Rodriguez and Luke Shaw.
Yaya Toure, Steven Gerrard and Eden Hazard are the only other midfielders to have scored more league goals than Lallana this term, while Aaron Ramsey and Raheem Sterling have also scored nine.
Southampton have insisted that they have received no official bid for the player from Liverpool.
Ahead of Southampton's final game of the season with Manchester United on Sunday, Saints coach Mauricio Pochettino said he was only aware of what has been reported in the media.
"I pretty much have the same information you have. The club has not spoken to me officially about anything (regarding Adam)," he said.
Sky sources suggest that Liverpool boss Brendan Rodgers is determined to complete a deal to sign Lallana before England depart for Brazil on June 8.
England coach Roy Hodgson will unveil his provisional England World Cup squad on Monday. Find out who makes the cut live on Sky Sports News and via the Sky Sports Today live blog.
Would you take Lallana to Brazil? Don't forget you can put yourself in Hodgson's shoes with Sky Sports' England Squad Selector. Click here to make your choices.