In 1995, the government increased the pension age for women from 60 to 65, phasing in the change between 2010 and 2020.
The coalition government of 2010 opted to speed up the process, bringing forward the qualifying age of 65 to 2018.
The Waspi group said many of the affected women had made financial plans based on the old state pension age and some retired before realising they could not claim their pension.
It previously called for impacted women to be paid up to £10,000 each in compensation.
An independent government review in March recommended compensation of between £1,000 and £2,950 for each of those affected.
Burnham said: “That was a report that all political parties should have accepted and acted upon.”
Starmer said the Treasury could not afford to pay out compensation, with Labour ministers warning the total cost could have risen to more than £10bn.
He added that “90% of those impacted knew about the changes that were taking place”.
But Burnham said the compensation “doesn’t have to be paid all at once”, and could be done through a “top up of the pension for a few years”.
The has contacted the government for further response.
