JD Sports Fashion and Sports Direct are competing with Halfords to take control of Evans Cycles, Sky News reports.
Evans Cycles is said to need £10 million to see it through the next few months, amid difficult high street trading conditions.
According to Sky News, JD Sports Fashion and Sports Direct International both tabled indicative offers last week.
The news has emerged just days after it emerged that Halfords was among bidders for the century-old retailer.
It is reported that JD Sports, which owns the Go Outdoors chain, is understood to have a genuine interest in buying Evans but it is not clear whether Sports Direct is committed to progressing further in the auction process.
Evans Cycles, which is owned by the private equity firm ECI Partners, is being advised by PricewaterhouseCoopers (PwC).
AlixPartners is understood to be advising the retailer's principal lenders.
The chain trades from just over 60 stores across the UK and traces its history to 1921, when the first FW Evans Cycles shop opened in southeast London.