Forward Paul Mullin posted pictures of the boots, taken at the club's Racecourse Ground, on Instagram on Monday.
Promotion-chasing Wrexham, owned by Hollywood stars Ryan Reynolds and Rob McElhenney, said the photographs were taken without the club's "knowledge or approval" and that they take a "neutral position" on political matters.
The club described the post as an "unwelcome distraction" before their home game against Halifax Town later on Tuesday but said the forward, who has scored 11 times in 14 league games this season, will remain a key player in their promotion bid.
"For the record, the pictures wouldn't have been permitted to be taken, had we known, and the issue will be dealt with privately by the club," Wrexham said in the statement.
"The club also acknowledge that everyone is entitled to their own opinion, whether that be employees or supporters, but would also highlight that an individual view cannot be fairly attributed as the view of everyone or the club itself.
"There is no more prominent example of this than the fact that the MP for the constituency within which the club is found is a Conservative seat."
Rishi Sunak formally became Britain's third prime minister in two months on Tuesday, taking over after predecessor Liz Truss' 44-day tenure at 10 Downing Street ended.
