Thousands of elderly homeowners could be entitled to property-tax breaks totaling $21 million over the next four years under a bill that won tentative approval from the D.C. Council on Tuesday.

But some lawmakers expressed concerns, with several suggesting ­changes before the bill is made final. Meanwhile, a broader property-tax relief measure was postponed amid concerns that it would unfairly benefit the city’s wealthiest residents.

The latter bill, introduced by Finance and Revenue Chairman Jack Evans (D-Ward 2), would limit yearly increase in property tax bills to 5 percent rather than the current 10 percent. It would also eliminate a requirement that homeowners pay tax on at least 40 percent of their home’s assessed value.

Evans agreed to postpone the council’s consideration of his bill until its next legislative meeting, with several members noting that a blue-ribbon commission recently considered and rejected making such a change.

The D.C. Fiscal Policy Institute, a liberal think tank, raised objections about the Evans bill, saying it would provide two-thirds of its relief — $32.5 million over four years — to the most valuable one-third of District homes.

Among the critics was council member Mary M. Cheh (D-Ward 3), who represents the city’s most affluent ward. She said it would be a “farce” for the council to tighten the cap after the Tax Revision Commission declined to recommend doing so — even if her ward stands to most benefit.

“My constituents, I think, are also people of fairness and equity,” she said.

Yvette M. Alexander (D-Ward 7), who represents a less-well-off area of the city, also said she had concerns about the Evans bill. But she backed the other measure, introduced by Anita Bonds (D-At Large), to exempt homeowners 75 and older from property taxes if they have lived in the city for at least 15 consecutive years.

“Seniors, yes,” Alexander said. “Rich people, get over it.”

In a letter to council members, Mayor Vincent C. Gray (D) urged the council to vote down Evans’s bill, citing its “negative financial impact on the District’s revenues, its inequitable treatment of District homeowners and . . . because it does not increase the District’s competitiveness regionally.”

Gray did not take a position on Bonds’s bill. Some council members expressed concerns that the measure offered no relief to renters and urged changes before a final vote, but the measure won unanimous support in an initial vote.

Jim Graham (D-Ward 1) said the bill would help fixed-income seniors deal with rapidly rising home values. “We don’t want them to sell their houses,” he said. “We want them to stay in our community, where they nurture and enhance our diversity.”

In other business Tuesday, council members gave unanimous approval to a billion-dollar plan to bury some electric lines in a bid to improve reliability during severe weather.

Under the plan, Pepco and the city would together work to put 60 of the city's most troublesome power lines underground. The costs would be borne mainly by Pepco rate-payers, via a surcharge on monthly bills.