It also includes the construction of a second Mt Victoria tunnel, but not until at least 2030.
Foster said it was clear from government advice that the plan was not ready to be released when it was, adding that council held "a few workshops" before being asked to rubber stamp the plan.
"In theory we were going to have three minutes to discuss the biggest investment in Wellington's history," Foster said.
"To me, that was not adequate."
Foster, whose campaign for the mayoralty was based in large part around his support for the second Mt Victoria tunnel, told reporters he would seek to renegotiate aspects of the plan.
Twyford has previously played down the chances of a renegotiation, but Foster said there had "certainly [been] some indication" from the Transport Minister's office of a willingness to discuss details.
While Twyford has said repeatedly that he had ruled out congestion charging, Foster said he hoped to revisit the issue because ratepayers would not be able to afford the city's contribution to the project.
Foster told the audience that existing Wellington ratepayers were forecast to face rates increases of more than 50 per cent over the next decade.
"We're being asked to spend $1.2 billion in today's money, local contribution, of which 65 per cent is theoretically coming from rates," Foster said.
"You put that on top of our rates bill at the moment, ratepayers simply won't be able to afford it."
In a statement, Twyford confirmed he had invited Foster and Greater Wellington Regional Council chairman Darren Ponter "to come in a have a chat tomorrow about how we can move forward with Let's Get Welly Moving".
Twyford said the plan was the result of "years of collaboration" between councils and central Government.
"I'm happy to talk about the details of the implementation of the projects, which were agreed by all three parties including Cabinet, but the fundamentals are not up for renegotiation."
His office declined to comment on whether Twyford believed the timing of a second Mount Victoria tunnel was a "detail" or a "fundamental".
Twyford said he had "already ruled out congestion charging for Wellington, as did former mayor Justin Lester".