Ramsdale only moved to Southampton last summer from Arsenal, but is now set to leave St Mary's as he has no interest in playing in the Championship next season.
West Ham United are said to have already made contact, but The Sun says United are also keen.
United are eager to improve their goalkeeping stocks with fresh doubts emerging over senior pair Andre Onana and Altay Bayinder.
Ramsdale, 26, favours staying in England, rather than leaving for abroad this summer.