The battle for Macy's (M) future is heating up as a private equity firm's quest to take over the American classic continues.

On Thursday, Arkhouse Management and its partner Brigade said in an SEC filing that they're working on a confidentiality agreement with Macy's that would allow the firm to conduct due diligence. Arkhouse, who is a major shareholder at Macy's, first announced a $5.8 billion offer for the company in early December, then upped its bid to $6.6 billion on March 3.

While Macy's rejected the December bid — citing concerns with financing — the department store giant seems more willing to deal this time around. According to the SEC filing, Macy's board responded to Arkhouse on March 11, indicating that they find the offer "less than compelling" and would not be willing to "transact at this price level."

However, the letter said that the board was willing to proceed with negotiating a confidentiality agreement that would provide Arkhouse access to Macy's books. Arkhouse has said in the past that it may be willing to raise its offer if it can gain more financial information, a request that Macy's has denied until now.

Yet it's unclear if the increasing urgency from Arkhouse is pushing Macy's towards a deal. Prior to the latest filing, Neil Saunders, GlobalData’s managing director of retail, told Yahoo Finance rumors are floating that "Macy's is still not impressed with this deal ... they believe the company is still worth more."

He added that others believe a sale to a private equity firm may be the "wrong trajectory" for the storied company, with some speculating that Arkhouse is primarily interested in cashing out on Macy's real estate holdings.

In late February, Arkhouse's managing partner Gavriel Kahane told Yahoo Finance the firm's focus is to give shareholders a premium — via a buyout — then lean into the "iconic" retail brand and real estate.

Though department store retail has been on a downslide, Macy's “should be growing market share in that troubled space,” he said. “There are a lot of companies that thrive in industries [or] sectors that have secular headwinds in them.”

In the filing, Arkhouse said that Macy's hasn't evolved with the times, calling it "an iconic American brand that has fallen on hard times in recent years" as online retail and consumer preferences shifted.

Macy's declined to comment to Yahoo Finance on the latest filing.

Arkhouse's March all-cash offer values Macy's at $24 per share — a 14.3% increase from its original proposal of $21 per share — and a 51.3% premium to Macy’s share price on Nov. 30, 2023.