Goodwillie played for the club in a friendly match earlier this month, but the club has not confirmed whether it has signed him.
BBC Scotland has contacted the club for a response to Aitken's ultimatum.
Goodwillie was found guilty of rape in a 2017 civil case and has never shown any remorse, according to Aitken.
She said that any club that signs him is making a clear statement about its attitude towards the safety of women and girls.
Rape Crisis Scotland has condemned the club's decision to play Goodwillie, saying that footballers are role models and it is not okay to have someone who has committed rape in this position.
Goodwillie's move to Raith Rovers in 2022 sparked a backlash, with sponsors threatening to pull out if the club signed him.
The club's officials met Goodwillie several weeks ago and decided to give him a chance.
They said that the player deserves to be left in peace and to be allowed to play the game he loves.
However, Aitken has said that she has asked officers to look at the council's agreement with Glasgow United and is ready for the city to walk away if necessary.
She said that it would be outrageous for any club to sign Goodwillie and that it would be at odds with how community facilities should be run.
Goodwillie's former club, Dundee United, has also distanced itself from him, saying that it has a zero-tolerance policy on any form of abusive or violent behavior.
The club said that it was disappointed to hear that Goodwillie was playing football again and that it would be inappropriate to comment further.
Goodwillie played for Scotland between 2010 and 2011 and has also had spells with Motherwell, Heart of Midlothian, and English sides Plymouth Argyle and MK Dons.
However, his career has been affected by the rape accusation, and he has had several moves to clubs in Scotland, England, and Australia collapse due to fan outrage.
Scottish football club Raith Rovers sparked outrage in January 2022 when they signed former Celtic and Aberdeen player Chris Goodwillie, who had been accused of sexual assault in 2018.
The move was met with condemnation from Rape Crisis Scotland, who described it as a "clear message of disregard to survivors of rape and sexual violence".
However, the loan move to Clyde collapsed and Goodwillie never played a game for Raith Rovers.
In September 2022, the club released him, admitting they had "got it wrong" by signing him.
Fast forward to February 2023, and Northern Premier League side Radcliffe FC based in Bury, Greater Manchester, found themselves at the center of a similar controversy after signing Goodwillie.
The move was met with immediate backlash, with many calling for the club to cancel the contract.
Four months later, in June 2023, Goodwillie's contract with Australian semi-professional club Sorrento FC was rescinded.
The club issued an apology to anyone "that may have been caused offence by his signing".
It is concerning that a player with a history of sexual assault accusations continues to find employment in the football industry.
The repeated signing and subsequent release of Goodwillie highlights the need for clubs to thoroughly vet their players and take into account their past behavior.
The controversy surrounding Goodwillie's signings also sheds light on the importance of addressing issues of sexual violence and supporting survivors.
The football community has a responsibility to create a safe and inclusive environment for all, and this starts with holding players accountable for their actions.
LONDON DAILY ARCHIVE
Glasgow City Council Leader Warns Football Club Over Signing of Convicted Rapist David Goodwillie
Glasgow City Council leader Susan Aitken has warned Glasgow United FC that it could be barred from playing at its facility if it signs convicted rapist David Goodwillie.
24 July 2023 · 02:47 UTC
