The Saints have secured the Polish international to a new three-year contract.
He will be a Saint until 2027, but they have not been able to convince Kyle Walker-Peters to renew his deal.
Martin told the Daily Echo: "He won't sign a new contract right now. I don't know if that will change at any point.
"It is a situation we have to be conscious and aware of. If it's not going to change then he needs to give us everything on the pitch as Stuey (Armstrong) and Che (Adams) did last year.
"It will be the owner's call in January (if Saints will cash in on Walker-Peters), it will depend on how much he has played by then and his level of performance.
"It will come down to how he is feeling and how we are feeling. It depends on so much. I can't give you a definitive answer on that.
"It also depends on who comes in for him and what clubs value him at because nobody has done that in the last three transfer windows.
"Kyle has been brilliant for us, and whilst that continues to be the case, it is not a problem at all."
"I think it's massive for the club," he added on Bednarek.
"Janny has grown into the season nicely. He's getting better and better.
"He's an outstanding person who has grown into a leader in the last year. With the responsibility he takes on and off the pitch, he's a big character for us.
"He's an amazing professional. It's been a privilege to watch him evolve over the last year. I don't think we should take him for granted. I think he's a top player.
"With a year left on his contract, it would have been easy for him to say I want to see that out. That has been other people's prerogative last season, and this season still.
"But he loves it here and he loves being a part of it. I think he likes working with us and has a lot of trust in us.
"It's a brilliant statement of intent from the ownership group to back us on that by giving Janny a new contract. I think he would have had a lot of offers if he allowed his contract to run out."