The northern bottle-nosed whale was in the water close to London's Prince Albert Bridge on Saturday after it had spent a day flailing upstream past historic landmarks such as Big Ben, the Houses of Parliament and the London Eye.
The Port of London Authority's Martin Garside, on a craft alongside the whale, said a rescue operation might be needed to get the animal back to the sea after it appeared to make no attempts to do so on its own.
"In the 24 hours since we've seen it, it has gone the wrong way. It's even further from the sea than it was yesterday," he said.
Rescuers passed an inflatable pontoon underneath the whale. The whale could now be hauled onto a vessel and carried to the sea, or loaded onto a lorry and taken by road.
The five metre long whale is the first of its kind to be sited in the river since 1913 and normally inhabits the icy waters of the North Atlantic Ocean way to the north.
Its odyssey up the Thames has fascinated Londoners and the UK in general. Hundreds of onlookers and TV crews flocked to the river bank for a glimpse of the whale. The interest was so great that police were forced to move people from the shore.
The whales are normally found
near Greenland and Canada
The fervour was such that at one point the UK's two leading 24 hour news channels interrupted regular programming and were running live coverage of the whale. The rescue efforts were also broadcast live.
Witnesses reported seeing injuries to the mammal, claiming its snout was bloodied.
Describing the discomfort the whale could possibly be in, Laila Sadler, scientific officer at the Royal Society for the Prevention of Cruelty to Animals, said: "A whale in the shallow water of the River Thames is like a human lost in the heat of the Sahara desert."
She estimated it could survive only for 24 to 48 hours in a river that has an average depth of between 8 metres and 6 metres.
"It also seems to be in distress, it has made two seemingly deliberate attempts to beach itself," she said.
When sick, old or injured, whales often get disoriented and swim off from their pod, said Mark Simmonds, science director at Whale and Dolphin Conservation Society, although witnesses reported seeing a second whale in a different section of the river on Friday.
The northern bottle-nosed whale is normally found in North Atlantic waters, off the coast of countries such as Greenland and Canada.
They prefer deep water, rarely being found in water less than 800-1000 metres deep, rather than the Thames which has an average depth of 6-8 metres.
Their main source of food is the Arctic squid.
They can reach lengths of eight metres – the length of a traditional London double-decker bus.
The whale is the first of its kind sighted in the Thames since 1913.
Scientists have said fluctuating ocean temperatures, predators, lack of food and even sonar from ships can send whales into waters that are dangerous for the mammals.
"It's extremely rare for one to turn up in a river in the United Kingdom. I suspect that the animal may be in ill health," said Tony Martin, a senior scientist with the British Antarctic Survey.
"It's going to be very confused. It's already stranded twice. The poor creature doesn't know where to go."
A fascinated British public, many of whom called radio and television stations offering to help, are hoping the whale returns to its natural home safely.