This year there were more than 30,000 applicants for 10,000 jobs at this stage, although some will have been doctors from abroad.
In the letter to the BMA Streeting said: "The choice is clear. You can continue to pursue unnecessary strike action, which will cause disruption to patients, harm the NHS's recovery and mean that at least some parts of this offer become unaffordable.
"Or you can put an end to this damaging period of industrial action and work in partnership with the government to both deliver real change and improvements."
The letter said iafter Thursday the NHS would have to start cancelling treatments and bookings ahead of the next walkout.
The offer has been made after months of dialogue between the union and government, which began in July after the last round of strikes.
Streeting has maintained he would not negotiate on pay after resident doctors had received pay rises totalling nearly 30% in the past three years.
But the BMA has argued that, despite the pay rises, resident doctors' pay is still a fifth lower than it was in 2008, once inflation is taken into account.
Responding to the offer, Dr Jack Fletcher, chairman of the BMA's resident doctors committee, said it "does not go far enough".
He said even with the expansion of training places resident doctors would still be left without a job at a crucial point of their training.
"We have also been clear with the government that they can call off strikes for years if they're willing to offer a multi-year pay deal that restores pay over time.
"Sadly, even after promising a journey to fair pay, Mr Streeting is still unwilling to move."
The strike next week is expected to cause significant disruption, particularly in hospitals.
Resident doctors represent nearly half the medical workforce and range from doctors fresh out of university through to those with up to a decade of experience.
They will walk out of both emergency and routine care with senior doctors brought in to provide cover.
While the NHS attempted to keep as many routine services running as possible during the last strike, thousands of operations and appointments still had to be postponed.