Potter is the manager of Swedish outfit Ostersunds, where he has won three promotions in five years to reach the top flight for the first time in the club's history.
According to the Daily Mail, Swansea officials have asked Ostersunds about Potter's availability to succeed Monk, who was sacked on Wednesday after one win in his last 11 matches.
But who exactly is emerging outsider Potter? And what's his background? Here, we take a look at the 40-year-old's credentials…
Potter was born in Manchester and started an unspectacular playing career as a trainee at Birmingham City. The young full-back broke into the first team at St Andrews in the early 90s before joining Stoke City in 1994.
After three seasons at Stoke he joined Southampton, where he made a handful of appearances in the Premier League under Graeme Souness before dropping down to the second tier with West Brom. He later spent time with Northampton Town, Reading, York City, Boston United and Macclesfield Town, but in 2006 he decided to hang up his boots at the age of just 30.
"I wasn't enjoying it as much as I'd liked," he explained in an interview with the Daily Telegraph last year. "I was only picking up short-term contracts in the lower leagues. I thought that rather than the game kick me out, I'd be proactive about it." In total, Potter had made close to 400 appearances across at 10 different clubs. Earlier in his career, he had also been capped by England's U21s.
Move into coaching
Potter worked as a football coaching manager at Leeds Metropolitan and Hull Universities and completed a masters in leadership and emotional intelligence before he was introduced to the chairman of Ostersunds, who offered him the job as manager in 2011.
The club had just been relegated to Sweden's fourth tier, but Potter turned their fortunes around and steered them to back-to-back promotions. "It's a really good football project I have here," he said last year. "To have this last three years in professional football in a different country has given me more time to develop a managerial style. Managers need to be given time to be successful."
Potter further enhanced his reputation by sealing their promotion to Sweden's top flight in November, with the new season set to begin in April 2016. Malmo, who finished bottom of Real Madrid's Champions League group, are also said to have expressed an interest in Potter.
Potter is not the first British coach to start his career in Sweden. Back in 1976, England manager Roy Hodgson took his first job in management at Halmstads, where he upset the odds to win two league titles in four years and launch a distinguished career.
Hodgson and Potter are not the only English coaches to enjoy success there, either. While Hodgson was in charge at Halmstads, Bob Houghton enjoyed similar success at Malmo, winning three titles in six years. Houghton did not enjoy the same level of success as his counterpart in the years that followed, but his coaching career took him around the world.
The pair were known as English Roy and English Bob in Sweden. English Graham will be hoping to follow in their footsteps.