The Premier League new-boys have money to spend following promotion to the top-flight but will not gamble by over-spending this summer.
And Brendan Rodgers, the Swans boss, remains keen to land either of Leroy Lita or Marvin Emnes before the start of the new season.
Mowbray is keen to keep both players at the Riverside Stadium but will be forced to listen to offers if Boro fail to move on the players he is looking to move on.
Swansea are thought to have been in touch with Middlesbrough. However, Rodgers - who recently signed former Boro striker Danny Graham - was unsuccessful in his attempts.
The Swansea manager would prefer to sign Emnes, but Boro are unlikely to even consider selling unless it looked like they would recoup more than the ?3.2m paid to Sparta Rotterdam for him in 2009.
Mowbray has been hugely impressed by the Dutchman and recently hinted that he would be looking to get the player to sign a new contract.
Emnes impressed during a loan stint at the Liberty Stadium last season and Rodgers has made no secret of his hopes of taking him back on a permanent basis.
Mowbray had been hoping for greater progress on his summer plans to reduce the club's ?20m wage bill by now.
But at this stage the likes of Kris Boyd and Didier Digard are expecting to be back at pre-season training after failing to seal full-time switches after their loans with Nottingham Forest and Nice ended.
Source: Northern Echo
Source: Northern Echo