After traveling 2 1/2 years without major mishap, crossing the Sahara Desert, the Antarctic and the Pacific Ocean, the amateur explorers of the Transglobe Expedition have run into an unexpected problem--the North Pole is melting.

More than 350 miles from the pole, the last major objective on their attempt at an unprecedented round-the-world trip via it and the South Pole, expedition leader Sir Ranulph Fiennes, 38, and Charles Burton, 40, were trekking across an ice floe today by foot and snowmobile, surrounded by water after a freakish heat wave broke up the thick icecap that normally covers the area this time of year.

"We will carry on," Burton told his wife Thelma over a crackly radio-telephone hookup to London.

"They will stay there and stick it out," she said.

At Transglobe's London headquarters, spokesman Anthony Preston said it was uncertain whether the two men were safe or how much progress they would make if the ice kept melting. He said the temperature had risen abnormally from about 20 degrees below zero to just a few degrees below freezing.

Earlier, Fiennes narrowly escaped death when his snowmobile plunged through thin ice and sank with his supplies, according to information relayed by his wife Lady Virginia from the expedition's Alert base camp in northern Canada.

Then the two were stranded for a day on a small ice floe. This morning it bumped into the much larger floe and they managed to hop aboard.

"If they reach the North Pole, and it is too late to continue from there because of the ice breaking up," Preston said, "they have every intention of returning and spending the next winter on the ice" to finish the trip.

The De Havilland twin-engined Otter that airlifts supplies from the camp to the two adventurers was repaired today after being damaged in a blizzard. It may be able to provide aerial reconnaissance of possible routes ahead on what remains of the icecap.

Two weeks ago, a fire in a storage shed at the camp destroyed other supplies, including snowmobiles and inflatable boats. Other snowmobiles were found and flown out to Fiennes and Burton. But replacements for the boats, which could have been used when the ice began melting, have not yet been obtained.

The explorers, who were seen off from Greenwich by their royal patron Prince Charles in September 1979, were scheduled to return this August. Preston said he believed commercial sponsors who have donated or paid for supplies and sophisticated equipment would continue their support if Fiennes and Burton were delayed.

Virginia Fiennes set up the radio-telephone hookup last night to boost the men's morale. Burton's wife described it as "the most emotional moment of my life. I just simply was not expecting to hear his her husband's voice."

"I am quite convinced they will get out of this," Mrs. Burton said. "Naturally I worry for their safety. I just tell myself they're going to get through."

She said she distinctly heard her husband say during the often unintelligible conversation, "See you in Greenwich."