McCormick, now 28, was jailed for seven years and four months in 2008 for killing two young boys in a car crash while drink-driving.
The former Plymouth goalkeeper is expected to be freed from Leyhill Prison next month and Swindon have offered him a trial, with a view to winning a permanent contract.
But the offer has angered the family of Arron Peak, 10, and his brother Ben, eight, who died in the collision on the M6 in Staffordshire in June 2008.
The boys' mother, Amanda Peak, told the Sun that Swindon's decision "was like a kick in the stomach".
"Swindon might be a family club but now they are hiring a man who has torn my family apart," she told the newspaper.
Wray, Swindon's interim chairman, who said McCormick had been training on day-release with the club since January, today said he would resign if the decision damaged the club's reputation.
He told BBC Wiltshire: "I'm not going to jeopardise the club's position on this.
"I'm not going to allow the reputation of the club, that we've worked so hard to build up, to be hurt by this.
"In terms of a principle, I absolutely would resign."
Speaking earlier this week, Mr Wray said: "We thought long and hard about it, and it has to be made clear that everybody's thoughts can only go to the family of those two young boys and their father who was badly injured.
"Tragedy is a word used too often, but this was a tragedy and it can never be changed.
"As a club we did not go courting this - the opportunity was presented to us after I was contacted by his agent and he will spend pre-season training here at Swindon.
"We have had to look at it in a reasonable way and look to produce a positive outcome in horrible circumstances."
McCormick was twice the drink-driving limit when the crash happened as he was returning from former team-mate David Norris's wedding in Bolton.
Arron and Ben were travelling to Silverstone racetrack for a dream day out with their father Phil Peak and three friends when their Toyota Previa people carrier was hit by McCormick's black Range Rover at 5.44am on June 7 2008.
The goalkeeper had been seen drinking beer and downing shots of Sambuca at the wedding reception.
Stoke Crown Court, where McCormick admitted causing death by dangerous driving and drink-driving, was told police investigators estimated his Range Rover was travelling at around 97mph moments before the crash.
Mr Peak, from Partington in Manchester, was at the wheel of the Toyota and suffered serious injuries that left him in a wheelchair.
Plymouth terminated McCormick's contract by mutual consent after he was charged.