The 27-year-old woman and her five-year-old daughter were killed instantly by the impact at Southall, west London on Wednesday.
The woman's 23-month-old son suffered multiple injuries and died later in Ealing hospital.
The driver of the Heathrow Express train is being treated for shock.
The Daily Mirror said he told accident investigators he knew the woman was going to jump.
"I was approaching the station at 100mph when I saw these people on the platform," he said.
"I knew the person was going to jump and put on the brake, but I wasn't able to stop in time."
Several people are believed to have witnessed the incident.
Police are examining CCTV of the platform and film from cameras believed to have been fitted to the front of the train.
Tapes from a newsagent at the station are also being checked.
A British Transport Police spokesman said officers are not looking for anyone else in connection with the deaths.
That poor driver and those poor poor children