According to the Daily Star, the Anfield club have made a £15m opening offer for the Germany international, who is currently with Joachim Low's side in Brazil for the World Cup.
Khedira, who missed much of last season with a cruciate ligament injury, is a target for many of the Premier League big hitters, including Arsenal, Chelsea and United.
Along with Liverpool, the quartet have reportedly sounded out Real Madrid's willingness to sell the 27-year-old. United were thought to be the front runners to sign Khedira, with central midfield an area in need of urgent reinforcement at Old Trafford.
New United boss Louis Van Gaal has been linked with a number of central midfielders, including Khedira, Kevin Strootman and Toni Kroos.
Chelsea meanwhile, recently completed the signing of former-Barcelona playmaker Cesc Fabregas. The report claims that all but rules them out of the race for Khedira, but Jose Mourinho may feel he still needs another holding midfielder.
Fabregas operates further up the pitch, and Mourinho only has Nemanja Matic if he continues to sideline John Obi Mikel. Like Fabregas, Brazilian international Ramires is much better going forward.
As for Arsenal, the Gunners have lacked another holding option since Alex Song departed for Barcelona. The Cameroonian international has been linked with a return, but Khedira would be the better, albeit more expensive, option.
Liverpool lead the race
But all three contenders are now trailing Liverpool, who are extremely active in the transfer market already. Brendan Rodgers reportedly wants to bring in two more Southampton players, Adam Lallana and Dejan Lovren, while Bayern Munich's Xherdan Shaqiri is also on his radar.
With Champions League to contend with next season, Rodgers wants to improve the overall strength in depth of his squad. He was forced to rely on loan signings last year, but is eager to get permanent deals done before pre-season.
Khedira featured in Germany's opening World Cup match, a comprehensive 4-0 win over Portugal. They face Ghana on Saturday in their second group stage match.
Sami Khedira - What are you getting?