United boss Louis van Gaal
is trying to strengthen his squad after they finished fourth in the Premier League.
Germany international Schweinsteiger, 30, and 29-year-old Spain player Ramos are among those who interest him.
Ramos' contract with Real expires in 2017, while Schweinsteiger's Bayern deal runs until 2016.
Ramos has made 445 appearances for the club since joining from Sevilla in 2005.
He has 128 Spain caps and was in the starting line-up as they won Euro 2008, Euro 2012 and the 2010 World Cup.
Schweinsteiger also has a World Cup winners' medal - starting for Germany in their victory over Argentina last year.
He has been with Bayern all his career and won his eighth league title in May. He also won the Champions League in 2013.
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