TfL's finances have been hit by the coronavirus pandemic, leading to it requiring a £1.8 billion Government bailout in November to keep services running until March this year.
Speaking on BBC's Politics London programme, Mr Byford said: "TfL absolutely will require additional subsidy for the next year and the year beyond that.
"Obviously, we will do our bit to cut our costs, we have in the past taken a billion pounds off the cost base, there are further savings baked into this financial sustainability plan.
"But the frank or stark reality is that without Government support, and with the chaos that Covid and the decimation on our finances that Covid has wrought, we absolutely will be needing financial support in the short term and we're making a very strong case to Government to achieve just that."
The Crossrail project was initially due to be completed in December 2018 but has been delayed and is over budget, with £825 million more Government funding given last month in the form of a loan.
Its budget was set at £15.9 billion in 2007 but it is now expected to cost around £18 billion.
Mr Byford said: "I've challenged my team to see if we can safely improve upon that first half of 2022 date, it has my personal attention.
"I'm not going to nail my colours to an earlier date until I'm certain that that can be achieved.
"We've had too many deadlines that have come and gone in the past, but I am driving the team to achieve the earliest, cheapest, safe possible opening date. So watch this space."
On the development of Crossrail 2, Mr Byford said "the case still remains" despite the project being mothballed as part of the November 2020 Government bailout.
Crossrail 2 would connect existing railway lines in Surrey and south-west London with stations in east London and Hertfordshire.
Mr Byford said: "We have prioritised more immediately important schemes but it's certainly a scheme that we still want to do.
"My view would be that the case still remains obviously, there's some uncertainty now with Covid."