Joe Biden made the short but significant journey from the White House to Capitol Hill on Wednesday for a key meeting to bolster Senate Democrats’ $3.5tn “human infrastructure” plan.

The president put in an appearance at the Senate Democratic caucus’s weekly policy lunch, one day after the majority leader, Chuck Schumer, announced an agreement on the spending proposal.

Biden posed with Schumer as they made matching gestures of air-punching resolution, as Biden said: “We’re going to get this done.”

Meanwhile the House speaker, Nancy Pelosi, sent out a letter to fellow House Democrats saying the plan indicates that an agreement on a final bill “will contain many of House Democrats’ top priorities, including transformative action on the investments needed to confront the climate crisis”.

Pelosi added: “This budget agreement is a victory for the American people, making historic, once-in-a-generation progress for families across the nation.”

Biden joined Senate Democrats for the closed-door lunch where he sought their support and discussed strategy for passing both a $1.2tn bipartisan infrastructure deal to rebuild America’s roads and bridges, and the larger Democratic package that also addresses environmental measures and the need for stronger social services.

Republicans voiced immediate objections to the plan’s massive size, as did at least one key moderate Democrat whose support would be critical to passage.

Biden urged senators to think about how the package would affect average Americans, said Senator Chris Murphy.

*He just kept on telling us to think about his neighbors in Scranton,” Murphy said, referring to Biden’s Pennsylvania home town. “Think about whether what we’re doing is going to pass muster with the folks that he grew up with.“

An Ipsos poll conducted this month for Reuters found that most Americans want the kind of infrastructure improvements that are included in the Biden plan.

It also found that nearly two-thirds of the country supports increasing taxes on “the highest-earning Americans” to pay for the improvements.

In response to reporters’ questions shouted to him as he left the meeting on Capitol Hill, Biden said: “Great to be home. Great to be back with my colleagues. I think we’re going to get a lot done.”

Biden represented Delaware in the US Senate for more than 35 years and served as Barack Obama’s vice-president for the Democrat’s two terms in the White House, before becoming president himself by beating Donald Trump in the 2020 election.

Biden later met with a bipartisan group of mayors and governors at the White House to discuss the $1.2tn bipartisan infrastructure framework.

Democrats face a tricky path ahead in getting the two measures approved by a narrowly divided Congress.

At the least they will need the support of all 50 of their senators – plus Kamala Harris’s tie-breaking Senate vote – to pass the $3.5tn package over Republican opposition in the 100-seat Senate, using a maneuver called reconciliation that can facilitate financing-related bills by getting around around the congressional chamber’s normal 60-vote filibuster threshold to pass legislation.

While some of the more liberal Democrats on Wednesday said they had hoped for a bigger plan, they had yet to reject the $3.5tn deal.

The Senate Republican leader, Mitch McConnell, wasted no time in going on the attack, saying that “with inflation raging” the plan “is wildly, wildly out of proportion to what the country needs right now”.

“With inflation raging … [the Democrats’ budget plan] is wildly, wildly out of proportion to what the country needs right now,” he told reporters.