States and localities have only distributed 11 percent of the tens of billions of dollars in federal rental assistance, the Treasury Department said Wednesday, the latest sign the program is struggling to reach the millions of tenants at risk of eviction.

The latest data shows that the pace of distribution increased in July over June and that nearly a million households have been helped.

But with the Supreme Court considering a challenge to the federal eviction moratorium, the concern is that a wave of evictions will happen before much of the assistance has been distributed. Some 3.5 million people in the U.S. as of Aug. 16 said they face eviction in the next two months, according to the U.S. Census Bureau’s Household Pulse Survey.

The Baker administration has said about $280 million in rental assistance has been distributed to landlords, out of $968 million in combined federal and state money allocated. Statistics released by Treasury show $144 million of the distributed funds came from federal sources, and went to 38,665 households through July 31. The city of Boston has distributed another $7.4 million to 5,798 households over the same period.

Treasury released additional guidance Wednesday to try to speed up the process. This includes programs to allow tenants to self-report their income and risk of becoming homeless among other criteria “when documentation is not available.” Many states and localities, fearing fraud, have measures in place that can take weeks to verify an applicant qualifies for help.

In Massachusetts, state statistics show half of all applications for rental aid are never finished, and 80 percent of applications are submitted incomplete. Landlord groups have blamed the lengthy application for scaring away tenants in need of aid, particularly undocumented ones, and progressives on Beacon Hill have garnered significant support for a bill that would both mandate reforms in the rental aid process while instituting a new eviction freeze through mid-June 2022.

Treasury also said states and localities now can distribute money in advance to landlords and utility providers “in anticipation of the full satisfaction of [the] application and documentation requirement.” And they approved providing money for tenants who have outstanding rental debt in collection, which would make it easier for them to find new housing.

“For those cities and states that wanted even more clarity that they can and should use simpler applications, speedier processes and a self-attestation option without needless delays – this answers that call,” said Gene Sperling, who is charged with overseeing implementation of President Joe Biden’s $1.9 trillion coronavirus rescue package.

“The guidance could not be more clear in expressing that this is a public health and eviction emergency that requires putting quick and sound rental relief above unnecessary paperwork that will not reach families in time.”

The administration also announced measures aimed at averting evictions at federally-back housing, including 400,000 rental units in Department of Agriculture-backed multifamily properties. It also is offering additional rental assistance to at-risk veterans and their families and working to ensure tenants in public housing can access rental assistance.

The news comes as anecdotal evidence and sales data for two- and three-family properties show landlords are facing significant financial struggles, and some may be abandoning the business altogether.