He said the bus fare cap is "a great measure for anyone not driving", but added that fixing the train fares would "help many more".
"Working in London I would expect the price of the train to go down because the commute is horrendously expensive nowadays," he said.
The new cap will take effect from 1 January 2027 for fares on participating buses outside of London.
The scheme is expected to cost more than £500m, which Burnham said would be paid for by "reprioritising" government funds.
But Shadow Chancellor Mel Stride said the government had not actually specified where the money would come from.
In its announcement, the government said £454 million will be reallocated from the Department for Energy Security and Net Zero (DESN). DESN said this will be used in addition to funds already earmarked by the Department for Transport for buses.
In January 2025, an England-wide bus fare cap rose to £3 under Sir Keir Starmer's Labour government, although Liverpool and Manchester are examples of city regions where fares have been held at £2.
While a lower fare cap is welcome, the Campaign for Better Transport said rural areas were still suffering from fewer bus routes and services, adding that such areas had been more affected by council cut backs in the last decade than urban zones.
Additional reporting by Adam Clarkson.