Directors at the online discounter met to discuss Mason's future at a board meeting in Chicago on Thursday.
Mason, chief executive of the online daily deal firm once billed as the fastest growing company ever, said ahead of the meeting that it would be "weird" for the company not to question his leadership after the collapse in Groupon's share price.
The discussion at Groupon's regularly scheduled meeting comes amid reports of discontent between Mason and co-founders Eric Lefkofsky and Brad Keywell.
According to venture capital sources Lefkofsky backed Mason at the meeting. All three founders have "super-voting" shares that carry 10 times the votes of ordinary shares making it all but impossible to dismiss Mason as long as he has the support of his fellow founders.
"We hear the meeting was very contentious. The independent directors want him out. He's in over his head. But basically you can't fire this guy without his own permission," said Sam Hamadeh, chief executive of the analyst PrivCo.
Paul Taaffe, company spokesman, said in a statement: "The board and the management team are focused on the performance of the company and they are all working together with heads down to achieve Groupon's objectives."
Mason, 32, founded Groupon in his twenties and built it into a web phenomenon that, according to Forbes magazine, was the fastest ever to record $1bn in sales.
Google offered $6bn for Groupon in 2010 but that was rejected in favour of an initial public offering (IPO). The share sale, in November 2011, was the largest tech IPO since Google and valued the the company at $13bn.
Since then, Groupon's decline has been dramatic, and it is now valued at less than $3bn. Shares that sold at $20 at the IPO have sunk as low as $2.60, although the share price rallied to over $4.40 on reports that Mason's days are numbered.
The reports about Mason's future first surfaced on the tech blog All Things D.
Speaking about the reports at a conference held by the business news website Business Insider on Wednesday, Mason said: "Here's a news flash: our stock is down about 80%, it would be weird if the board wasn't discussing whether I'm the right guy to do the job. It's their chief responsibility to ask that question.
"If I ever thought I wasn't the right guy for the job, I'd be the first person to fire myself."
The company has been hit by accounting irregularities, departures of senior staff and slowing sales, especially overseas. Recently the firm has diversified into new areas like Groupon Goods, which sells discounted items like heart rate monitors and yoghurt makers. Growth in that business has been good and Mason said this week was making $500m in revenues one year after its launch. But analysts point out that Groupon Goods has very small margins.
Groupon posted a loss of $2.98m in the third quarter and its shares hit new lows as it announced slowing revenue growth amid weakness in its European business.
"We have a North American business that's growing … that's really the model for what Groupon can be," Mason said Wednesday. "Because of our strategy to grow quickly [in Europe] and capture market share, we didn't invest in technology like we did in North America. We're paying for that now. We now have the playbook, it's the playbook for North America."
Mason's woes come amid signs of a major slump in the daily deal business model. LivingSocial, Groupon's chief rival, is reportedly preparing to lay off 9% of its staff. The firm is part-owned by Amazon, which invested $175m in the company in 2010. That investment was virtually wiped out last month when Amazon reported a third-quarter net loss of $274m; $169m of those losses were driven by an impairment charge from its stake in LivingSocial.
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