This is according to Spanish newspaper Sport, who claim United are the only club to have submitted a contract offer to the Brazil international.
French champions Paris Saint-Germain are also interested in the 31-year-old, according to recent reports, but as things stand only Manchester United have submitted a formal offer.
Van Gaal does not trust Rafael
This latest development further reinforces the feeling that United boss Louis van Gaal does not trust Rafael, who could leave Old Trafford this summer, with his contract set to enter its final 12 months.
The 24-year-old has endured an injury-hit campaign and has failed to win his place in United’s starting line-up back from Antonio Valencia, a makeshift right-back who has impressed over recent months, since returning to action.
Rafael has played just four minutes of Premier League football since New Year’s Day - and has not completed 90 minutes since the 1-1 draw with Chelsea back on October 26.
Barcelona unable to replace Alves until January
Alves, meanwhile, has been a regular fixture for Barcelona all season and the major surprise is that the Catalan giants are planning to offload him on a free transfer this summer.
The South American is still, by some considerable distance, the best right-back at Barcelona - neither Martin Montoya nor Douglas have convinced at the Camp Nou - and the leaders of La Liga are unable to replace the defender this summer because of their existing transfer ban.
While Barcelona are permitted to sign new players this summer, they are unable to register them until their transfer ban expires in January, so completing business at the end of the season will be extremely problematic for the club’s board.
The fact there are elections taking place at Barcelona, however, means potential presidential candidates are looking to deliver a stellar signing to arrive in the January transfer window - and the name on everyone’s lips at the moment is Juventus midfielder Paul Pogba.
The France international has also been linked with a return to Old Trafford, but the way Ander Herrera has been performing of late may convince Van Gaal against spending a record-breaking transfer fee on the 22-year-old.
Alves better than Rafael or Clyne?
A number of right-backs have been linked with Manchester United over recent months - including Southampton’s Nathaniel Clyne - but Van Gaal might favour a move for Alves with the Red Devils set to qualify for next season’s Champions League campaign.
The vastly-experienced Alves remains a very competent right-back, while his ability to bomb forwards and contribute in attack is still second-to-none.
Manchester United fans: would Dani Alves be an upgrade on Rafael? Tell us by leaving a comment in the box below!