The 37-year-old from El Salvador, who claims he was adrift for 13 months in a small fishing boat, had been expected to leave the Pacific outpost on Friday but medical staff advised against it, with Alvarenga still suffering dehydration.
Christian Clay Mendoza, the deputy chief of mission at the Mexican embassy in Manila who is acting on his behalf, said in Majuro that he needed more time to recover.
"We must consider his health, that he is fit to travel and get the doctor's recommendation. We're hoping in the next three or four days," he said.
Alvarenga has been treated at Majuro Hospital since his arrival from Ebon Atoll, where he washed ashore a week ago clad only in ragged underpants.
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He arrived at a press conference today clutching onto men on both sides and was helped up a flight of stairs and along a walkway to the venue.
"I want to thank the government of the Marshall Islands for all it has done and the friends who have helped me while here," he said in brief comments before being whisked away for more medical tests without taking questions.
"He's been through a lot and is still recovering but in good health," said the diplomat, who was jetted in from the Philippines when early reports said the castaway from Mexico.
It was later established that he was from El Salvador but had lived for years in Mexico and set sail from there on a fishing trip in late 2012.
He then said he became lost and drifted some 12,500 kilometres to the Marshalls in a small boat.
He said he survived by eating raw fish and birds as well as drinking turtle blood, urine and rainwater.
Experts say it is theoretically possible to survive such a journey.
Alvarenga set sail with a young companion he described as a teenager named Xiguel, who starved to death four months into the ordeal because he would vomit raw fish and birds and was thrown overboard.