Officers are treating the death as "unexplained" but have confirmed that the man had not been hit by a train.
Mr Speight, 42, disappeared last Monday, three months after the death of his fiancee, Natasha Collins, in the bath of the flat they shared in London.
Miss Collins' mother was devastated at the latest news, her spokeswoman said.
She asked for Carmen Collins to be given time to grieve in peace.
The BBC released a statement saying: "This is very sad news and our thoughts and sympathies are with Mark's family and friends.
"Mark was a hugely talented and very popular presenter for many years."
Video footage
The body of the man was found at about 1000 BST at the station.
Mr Speight's relatives have been informed of the discovery, police said, but a formal identification has yet to take place.
Video footage from CCTV cameras at the station is to be studied by officers as part of their investigation.
The presenter, who was the host of BBC children's show SMart until February, was reported missing last Monday when he failed to meet Miss Collins' mother as arranged.
He was recorded on a security camera entering Queen's Park underground station in north-west London.
Then, 22 minutes later, he boarded a train on the Bakerloo Line, which serves Paddington.
'Fragile state'
Relatives had made a number of appeals over the past few days, with his father, Oliver, urging him to get in touch and not to "give up" during his time of grief.
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Very sad event really, shame the lad couldn't cope with it all and probably took his own life.
