The Swans were prepared to pay £1m for the 24-year-old but have been unable to come to an agreement with the Championship side, according to Sky Sports News.
Despite Boro’s eagerness to keep the centre-back, boss Tony Mowbray has admitted that he may be resigned to losing his captain.
"Matthew is from a different timescale in terms of salary and sometimes, when players are running out of contract, we have to listen to offers," said Mowbray.
"If anything happens and that does go through, that will release some funds and we'll utilise the money in different areas of the pitch. I've said all along there may be outs to create ins.
"Matthew's got to get ready for Sunday now and if anything happens between now and then, fine.”
Middlesbrough face former Premier League side Birmingham City at the Riverside on Sunday.