In response to growing questions over the changes, Downing Street said it would provide "clarity" on how it would expand payments "as soon as we can".
Caroline Abrahams, charity director at Age UK, said: "We are encouraged by the chancellor's words, but will wait to see the detail, which needs to be published very soon if changes are to be made in time for this winter, something that is absolutely crucial if we're to protect the pensioners at greatest risk."
Earlier on Wednesday, pensions minister Torsten Bell said there would be no return to the previous system in which the payment was made to all 11.4 million pensioners.
"It's not a good idea that we have a system paying a few hundreds of pounds to millionaires, and so we're not going to be continuing with that," he said.
The government has not provided details of who would qualify as a millionaire.
Sir Keir told Parliament that the reason the government could expand the number of people receiving the payment was because Labour had "stabilised the economy".
Reeves said the economy was now in a "better shape," and that the government "had listened to the concerns people had about the level of the means test" in a speech in Greater Manchester announcing billions of pounds of investment in transport infrastructure in England.
But Daisy Cooper, treasury spokesperson for the Liberal Democrats, said pensioners deserved an apology from the government.
"This whole debacle has caused needless misery for millions of pensioners," she said.