The ex-footballer is working at Macclesfield at present, but may be in line for a step up.
Jon Brady resigned at the League One outfit Northampton, which is leading to a search for a new head coach.
Brady's assistant Ian Sampson is in temporary charge, but The Sun states he will not get the job permanently.
Savage is one who the club’s hierarchy admire and would like to see in charge.
A statement from Cobblers Chairman Kelvin Thomason said: "Upon receiving Jon’s resignation yesterday, the Board of Directors have discussed matters and given the circumstances have agreed to accept.
"We really appreciate what Jon has done for the club, initially within the academy and over the last four years or so as first team manager.
"I personally appreciated Jon's kind words earlier this week when he acknowledged how well supported he had been by the club.
"However we also all fully understand the real pressures that come with being a manager and while we all would have preferred things to be different, we understand Jon's decision.
"We will now work with Jon and the LMA on his exit agreement.
"In the meantime we are fortunate that we have a talented, passionate and experienced staff, headed up by Ian Sampson, to oversee things in the interim and allow us the time to work through matters.
"We also firmly believe we have both the talent and quality within the squad and every confidence in the players at the club.
"It is clear we are a bit of a wounded animal on the pitch at the moment, but we know that the Cobblers supporters will always step up when needed and are sure they will get behind the lads at Sixfields on Monday."
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