The Saints are considering a move for the Mexico international, who would cost around €19 million, but may face competition from Atletico Madrid
The Saints have renewed their interest in the Mexico international after attempting to sign the 26-year-old in January when the Red Devils informed them that 'Chicharito' he could go out on loan.
Then boss David Moyes subsequently pulled the plug on a potential move away, but new boss Louis van Gaal has refused to rule out an Old Trafford exit for the striker.
When asked after the friendly win in Washington whether he would consider selling Chicharito, the Dutch coach told reporters: "That I cannot answer because he is a player of Manchester United at this moment."
Southampton, meanwhile, are cash-rich having sold Luke Shaw, Adam Lallana, Dejan Lovren, Rickie Lambert and Calum Chambers and with Tottenham keen on Jay Rodriguez and Morgan Schneiderlin.
Manchester United are believed to want around €19 million for Chicharito, who joined Manchester United from Guadalajara in 2010 and has become a fan favourite at Old Trafford.
Southampton may face competition from abroad, however, with Spanish champions Atletico Madrid interested in the Mexico international and manager Diego Simeone admitting a move could happen.
"With Javier Hernandez and plenty of other great players, anything is possible. We can't close the door to anyone until the transfer window closes," he said.
Chicharito has also been linked with a move to Inter, where he would link up with former captain Nemanja Vidic, and admitted he was happy to hear of the Serie A side's interest.