The shirts have prompted complains from players from both the men's and women's players amid suggestions it's affecting the performances of the men's first team this season.
The Daily Telegraph reports talks have already taken place over the possibility of finding an amicable way out of Villa's 'multi-year' deal with British sports manufacturer Castore at the end of this season.
It adds that those talks are not necessarily as a direct result of the issue with this season's shirts. The Telegraph says Villa and Castore have declined to comment on the deal.
Castore earlier promised to "address this issue as quickly as possible to meet the standards we expect."
The Telegraph also says that as well as working on a long-term solution on Villa's shirts for the remainder of this season, Castore are looking for any short-term options for the men's and women's teams for this weekend's matches.
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A Castore spokesperson said: "There has been some media speculation about a potential issue in the football kit supplied by Castore to Aston Villa Football Club. We are working closely in collaboration with the club to address this issue as quickly as possible to meet the standards we expect. We would like to thank the club for their patience and support to date.
"As a proud new British brand, we always hold ourselves to the highest of standards and strive to do everything we can to constantly improve the performance of our products. This means addressing any customer concerns with promptness and humility."
Newcastle have already invoked an exit clause in their deal with Castore and their kits will be made by adidas from next season.
It's understood Villa co-owner Nassef Sawiris owns a stake in the German sports manufacturer, prompting speculation that Villa could follow suit with Newcastle.
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At the time of writing, Villa's kits for this season are not among those for sale on the Castore, JD Sports or Sports Direct websites, although it's not clear whether this is related to any talks over a potential early termination of their contract.
It was reported earlier this week that players from the Villans' men's and women's teams had raised complaints that this season's shirts quickly became wet through and cling to their bodies, thus affecting their performance.
One source close to a male Villa player said: "The players are having to play in soaking wet T-shirts and it is a problem that needs to be solved. It cannot go on all season. The players look like they've jumped in a swimming pool after about ten minutes."
Meanwhile, a source close to the women's team has been quoted describing the kit as 'really bad' to play in, while another separately voiced 'deep' concerns and is hoping for a solution.
Villa's men's team entertain Brighton on Saturday lunchtime, with the women's team kicking off their season against Manchester United on Sunday. Both games are being broadcast live on television.
On Villa's shirt issue with Castore, Andy Brian, commercial contracts lawyer and partner at law firm Gordons, said: "If the supply contract contains a detailed spec which is not being achieved, then it will be pretty straightforward to resolve; Castore will have to supply according to that spec.
"I don't know what this contract says, but perhaps the more likely scenario is that Castore is supplying products which do comply with the contract, but Aston Villa are not happy with the quality.
"Realistically, it will likely be resolved commercially, but it's pretty embarrassing."