The White House has said little about a fourth stimulus check. On May 4 during a press conference, White House Press Secretary Jen Psaki said "we'll see what members of Congress propose" when a reporter asked about another round of direct payments.
Although the post-pandemic economic recovery is ongoing, there's currently no legislation in Congress supporting a fourth stimulus check. There aren't even any specific stimulus-check guidelines on the table.
The most noteworthy economic effort in government currently is President Biden's $2.25 trillion infrastructure investment proposal, the American Jobs Plan. That proposal doesn't include a fourth stimulus check or anything like the direct payments that millions of people received over the course of the pandemic. Rather, it's a long-term initiative to rescue, recover and rebuild the country's financial standing.