WASHINGTON, June 1 (Reuters) - The United States is making progress towards the Fed's maximum employment and 2% inflation goals, but the depth of the remaining problem still requires the central bank not jump the gun in starting to tighten monetary policy, Fed Governor Lael Brainard said on Tuesday.

"We are seeing welcome progress, and I expect to see further progress in coming months," Brainard said in remarks prepared for delivery to the Economic Club of New York. But "jobs are down by between 8 and 10 million compared with the level we would have seen in the absence of the pandemic. And it will be important to see sustained progress on inflation," not just a temporary jump.

Some of the factors fueling current strong growth, including fiscal spending and the rush by households to take advantage of a broader economic reopening, are likely to fade over time, Brainard noted, another reason the Fed should not pull back too soon.

"Remaining steady in our outcomes-based approach during the transitory reopening surge will help ensure the economic momentum that will be needed," to make sure inflation hits and stays at the Fed's target, and people have as much time as possible to restart their old jobs or find new ones, she said.