Cirkin's future has been uncertain, with the left-back missing the first half of the Premier League season through injury and now into the last six months of his current deal.
But since returning from injury, Cirkin has impressed, taking his chance in the absence of Reinildo during the African Cup of Nations and featuring in Sunderland's last six Premier League games.
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Le Bris has previously refused to be drawn when quizzed on Cirkin's contract situation and future, and still says it's a "bit early" for a decision to be made.
But he has now strongly suggested that he'd like Cirkin to stay and believes a deal can be agreed, providing Cirkin continues to impress on the pitch and wants to stay.
"At the minute, I'm happy with Dennis, and he's doing well," said Le Bris.
"He had a long period without playing, so this period is really important for his consistency. It's more a step-by-step process at the minute, for him and for us.
"We trust each other, which is really important. If we go all the way, maybe we'll have another decision. But at the minute, nothing is done.
"I repeat, probably the main word with Dennis and us at the minute is consistency. Consistency and trust. If he stays consistent and feels that it's the right place to go further, I think we'll find an agreement. But it's a bit early at the moment."
Cirkin is in contention to start tomorrow night's game against Burnley as Le Bris looks for a reaction from his side after last week's West Ham disappointment.
Reinildo went back into the side as first choice left-back after his return from AFCON, but the Mozambique international had a nightmare at the London Stadium and was replaced by Cirkin at half-time.
Le Bris must now decide whether to stick with Reinildo against the Clarets or give Cirkin a chance from the off.