Over the weekend the U.S. president said he was feeling much better, and his doctors talked about an early discharge from hospital.
On Monday that was enough to give stocks a lift.
Japan's Nikkei index closed over 1.2 percent higher, with most indexes around the region also seeing gains.
In Europe the regional Stoxx 600 index then rose as much as 0.7% in early trade.
BP, Shell and other oil firms were among the big gainers.
They jumped as much as 2% following gains for oil prices.
Crude too was helped by the reduced anxiety over Trump and the U.S. election.
Traditional safe havens like gold and U.S. government bonds edged lower as investors moved money into riskier assets.
A flurry of dealmaking also helped markets.
Swiss duty-free operator Dufry surged as much as 12 percent after it said China's Alibaba was set to take a stake in it.
Spanish bank Unicaja jumped 6% after it said it was in talks to merge with local rival Liberbank.
But all eyes remain on the U.S. president, and there's still confusion over his exact condition.
Until that clears, the mood on markets may remain volatile.
