A deal would cost him £100 million and analysts believe it may be one of a number of acquisitions he is considering after selling £110m of shares in Sports Direct last week.
However, City sources say he may face competition from Halfords and the online cycling retailer Wiggle if he wants to buy Evans which has been owned by private equity group Active Capital since 2008. Canaccord Genuity is advising on a sale.
The Evans chain is firmly established in the growing cycling market, boosted by the success of sports stars such as Sir Chris Hoy, for whom it sells a range of his own designed bikes, and Sir Bradley Wiggins.
Mr Ashley, who also owns Newcastle United and is offering a further £10m loan to Rangers, has also been linked with a bid for Debenhams after developing a partnership with the department store chain.
Sports Direct said on Friday that it now owns 16.6% of Debenhams’ shares but Mr Ashley said Sports Direct does not intend to acquire the business.