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.
Scorer of Real's injury-time equaliser in their Champions League final victory over Atletico Madrid in 2014, 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.
Real's interest in goalkeeper David De Gea will be used by United in their attempt to sign Ramos.
Van Gaal said at the end of last season it would not be easy to keep De Gea - United's player of the year for the second successive year.
The 24-year-old Spain international has refused to extend his contract beyond 2016.
The Red Devils have been linked with several potential replacements during what Van Gaal said could be a "rough" summer for some of his players given the changes he anticipated.
"A club like Manchester United has to be like that," said Van Gaal in May. "When you purchase other players you get a stimulus."
Van Gaal is particularly keen to recruit a right-back, central defender, central midfielder and striker.
United, who have already bought PSV Eindhoven winger Memphis Depay for £31m, are also interested in Tottenham forward Harry Kane.
Barcelona full-back Dani Alves also interested them, but he has signed a new deal at the Nou Camp.

The watercolour, attributed to Queen Victoria's favourite artist Sir Edwin Landseer, was sold by JP Humbert of Whittlebury, Northamptonshire.
The painting has attracted both firm supporters and those who doubt whether it does depict the Brontes.
It was sold to a collector who plans to do more research and resell it.
Bidding took off just 15 minutes before the end of the "timed" online auction with the painting sold for £40,550 hammer price (£50,038 including buyers premium) to an private investor believed to be based in the UK.
Auctioneer Jonathan Humbert said: "We are very pleased our theory has been accepted and endorsed by the establishment.
"The evidence was compelling that this is the Brontes as painted by Landseer and its successful sale has proved that research and factual evidence will overcome apathy and negativity."
Mr Humbert had decided to pull the picture, which he believes to be of "national importance", from an auction in 2012 so more research could be done.
Landseer was a popular Victorian painter best known for his animal portraits and designing the bronze lions in London's Trafalgar Square.
The Bronte family moved to Haworth, West Yorkshire, in 1820 where the Reverend Patrick Bronte was appointed Curate of Haworth.
They lived at the Haworth Parsonage from 1820 to 1861, which is now the Bronte Parsonage Museum.
Manchester United are interested in signing Bayern Munich midfielder Bastian Schweinsteiger and Real Madrid defender Sergio Ramos.
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A painting claimed to be a previously unknown portrait of the three Bronte sisters has sold for more than £40,000 at auction.