Talks have continued with the 39-year-old this week as he continues to weigh up his next steps after leaving Rangers earlier this month.
Sunderland are one of a number of League One clubs to hold talks over the prospect of extending his playing career for another six months at least, with Charlton Athletic and Oxford United known to be actively monitoring the situation.
The Black Cats remain in the mix but there is not thought to be a decision imminent as of this morning.
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Johnson said his primary focus in the latter part of the week has been the League One clash with Bolton Wanderers on Saturday afternoon.
"There's no update on that one,” Johnson said.
"Talks have been going on but there's nothing to add at the moment.
"My focus has been on the game the last couple of days.
"Obviously the deadline for anyone in contract is Monday but out of contract it goes beyond that.
“I wouldn’t say we’ve put a deadline on it or anything like that, it’s just got to be right as we’ve said before that.
"The focus on my side has been training and making the new players feel as comfortable as possible.”
A new striker, whether it be Defoe or another player, is a key priority for Sunderland as they look to reduce the workload on Ross Stewart while Nathan Broadhead recovers from a hamstring injury.
The Black Cats signed Jack Clarke on loan from Tottenham Hotspur this week, primarily a winger but one who can play through the middle.