They have also said they wanted the government to address what they say is a shortage of jobs after the second year of training when resident doctors move into speciality training.
This year there were more than 30,000 applicants for 10,000 jobs at this stage, although some will have been doctors from abroad.
Streeting said the current government offer, which would provide additional training places and help with out-of-pocket expenses, would “be a win-win scenario where together we motor ahead with the progress we’re making in the NHS”.
But the BMA said the government had not presented “any proposals to us which will see the real change needed to fix the jobs crisis this year”.
It added that it had requested “as little as £1 per hour more over the next few years” in terms of a pay deal for resident doctors.
“Mr Streeting should be honest with patients: we are losing doctors to other countries and professions because they can’t find work in the UK – despite training here and wanting to work here,” the statement added.
Streeting also partly blamed previous industrial action for an increase in NHS waiting lists.
Latest figures show about 6.26 million patients were waiting for treatments to be carried out at the end of August, up from 6.25 million patients at the end of July.
“We had industrial action by resident doctors, that had an impact and set us back, as I feared it would and said it would at the time,” Streeting told the programme.
“We’ve also seen demand outstripping growth in activity.”
He added that he believed the NHS would “begin to see improvement again on waiting lists in the coming months.”