Bates is out of contract next summer and free to leave, but Mowbray does not want to allow his skipper to depart in a cut price deal and has told Swansea the situation.
The Boro boss, who is actively looking to recruit and is on the verge of signing Belgium international Faris Haroun after he passed a medical, delivered his warning to the Swans after last night's comprehensive 3-0 Carling Cup win at Walsall.
I have spoken to Brendan (Rodgers, the Swansea manager), admitted Mowbray. He has inquired. Matthew is our player. The bottom line is that Matthew is my captain, it would take a great offer for me to let my captain go. We will see.
Matthew has to get his head right. If there is a deal to be done it has to be right for our club. We are also aware of his contractual situation, in to his last year.
For the first time this summer Mowbray knows that whatever money comes in he can invest back in to his squad.
And as well as tying up a deal for Germinal Beerschot midfielder Haroun, who has returned to Belgium before completing his move to Teesside, Mowbray also has an interest in PSV Eindhoven's Orlando Engelaar.
The 32-year-old central midfielder, who can also operate at the back, is available for loan from PSV if the fee is right, although his wages could pose a problem.
We have a few things moving along, said Mowbray, who rubbished a link with Carlisle striker Francois Zoko. Engelaar is a good player at a top club, whether we can do something there I don't know.
One player Mowbray has already successfully got to sign a contract with Middlesbrough is Emnes and last night he showed why his boss wanted him so much.
The Dutchman scored all three goals the second a terrific volley to power Middlesbrough in to the second round of the Carling Cup at Walsall.
And Mowbray said: Emnes has the talent and technique to get the goals for us. We have to get him in the right positions to score for us and his second, even if it was a great ball from Justin Hoyte, was what we want to see more of.
He showed he has that technique to score goals and it's that sort of talent that makes the difference.
Now Mowbray would like to tie up the loose ends and tie up a new contract for Julio Arca, who yesterday revealed he was waiting for the manager to get back in touch.
Arca was quoted on Sky Sports' website saying: The pay cut is not a problem, I understand where the club is financially as Mr Gibson [Boro chairman, Steve] explained this to me personally.
"I accept the terms that the chairman offered to me but there was just one issue about the old contract which needed to be resolved.
My advisor met with the club on Saturday and they talked about a solution, I am flexible about this. I am now waiting for a call from the manager to conclude things, it's up to him now.
And Mowbray added: There's been a lot of talking and I am focusing on the team, I am not getting involved.
Source: Northern Echo
Source: Northern Echo