Iceland, the smallest country ever to qualify for the tournament, face hosts France in the quarter-finals on Sunday, after sending England packing in the previous round.
Frenchman Agnel set the world record for the 400m freestyle in 2012, but Iceland's coastline is more than 10,000 times longer than the 200 metres swims he specialises in these days.
"If the Icelanders win the Euro, I will swim a tour of Iceland," Agnel wrote on Twitter, prompting a series of comic responses.
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Icelandic coach Heimir Hallgrimsson was just one of many to relish the offer, warning the swimmer to "pick the right weather".
One Twitter user promised to welcome the Frenchman with hot chocolate at the finish line, while another tweeted: "Good luck swimming around Iceland. Warning it is ICE-COLD."
Sea temperatures in Iceland hover in and around the zero-degree mark for most of the year, rising above 10 degrees Celsius (50°F) in the summer months.
One user posted a map of Iceland's 2,008-kilometre (1247.7-mile) coastline and a rough calculation of how long the Olympian would take to complete the swim.
"28 days at 3km/h, sorry if I'm a bit off the mark," the tweet said.
An estimated 10 percent of Iceland’s 330,000 population is currently in France to watch their side, with more expected to travel to Paris for Sunday's match.
(Writing by Simon Jennings in Bengaluru)
