Ashley, the billionaire owner of Newcastle United soccer club, made his fortune from building retailer Sports Direct. More recently he has snapped up stakes in other chains hit by the rapid shift online and weak consumer spending.
He holds 30 percent of Debenhams, once Britain's biggest department store.
Late on Thursday, Ashley said he was willing to step down as CEO of Sports Direct if shareholders at Debenhams backed his push to remove all but one of the existing board members and install himself in an executive position.
Analysts said the 54-year-old was trying to take control of the company without buying the 70 percent he does not already own, and noted their surprise that Ashley was willing to step back from the top job at Sports Direct to do so.
He owns 61 percent of Sports Direct.
"If Mr Ashley were to be appointed to the board of directors of Debenhams during this business critical period ... Mr Ashley would carry out an executive role and would focus on the Debenhams business," Sports Direct said late on Thursday.
Shares in Debenhams, which have lost 97 percent of their value since a peak in 2015, jumped in early trading and were up 14 percent at 1050 GMT to 3.5 pence.
Ashley's request for an investor meeting comes only days after Debenhams, which dates back to 1778, warned on profit again, saying its restructuring was complicating its efforts to trade.
Debenhams said it was "disappointed" that Sports Direct wanted the board changes, adding that it has been engaging with stakeholders in its bid to restructure its debt and close stores to guarantee its survival.
Debenhams, a presence on most British high streets with its giant stores, has become vulnerable to a takeover since it failed to keep up with rivals in the shift to sell online.
Ashley has in recent years pounced on other weak retailers, including department store chain House of Fraser and analysts have speculated that he could put the two together.
At Debenhams he has already joined forces with another retail investor to oust the chairman and remove the CEO from the board, a highly unusual approach.
Analysts said Ashley now seemed "obsessed" with moving in to department store retailing, where he can use excess space to sell the brands that he already owns.
"If the vote passes, he would give up his role as CEO of Sports Direct, which is worth around 1.4 billion pounds, for an executive role at Debenhams, which is worth around 40 million pounds," Laith Khalaf at Hargreaves Lansdown said.
If Sports Direct's plan goes through, Chris Wootton, currently Sports Direct's deputy chief financial officer, would step up to take the role of acting chief executive.
Ashley founded Sports Direct, a discount sports goods and clothing retailer, in 1982 and has built it into one of Britain's most recognisable retail names.
Both he and the company hit the headlines in 2016 when he was called before a parliamentary committee to answer questions over work practices at the group after unions accused the company of creating a "culture of fear" for staff.
Ashley, who initially refused to attend, told the hearing he had been shocked by some of the accusations and admitted the company may have outgrown his ability to manage it.
During the hearing he said the company may have paid some warehouse staff below the statutory minimum wage by requiring them to queue for security checks on their own time. He also said allegations of employees being dismissed without notice and instances of sexual harassment were unacceptable if true.
Shares in Sports Direct were down about 1.1 percent.
(Additional reporting by Joe Green; editing by Jane Merriman)
By Kate Holton and Muvija M