BOSTON — The administrator of a fund created to help people injured in the Boston Marathon bombings and the families of those who died in the attack plans to meet with victims next week with hopes of cutting the first checks by the end of June.

Kenneth Feinberg is overseeing The One Fund Boston, which had taken in more than $28 million as of Friday and should get another significant boost from an upcoming benefit concert featuring performers such as Aerosmith and New Kids on the Block.

Feinberg, a Massachusetts native and Washington, D.C., attorney who also handled victims’ compensation after the Sept. 11, 2001, terrorist attacks, the BP oil spill and other calamities, planned to release a tentative proposal on Monday for disbursing money from The One Fund.

The proposal would be made in advance of two meetings with victims, one scheduled for that evening and another on Tuesday, at the Boston Public Library. The library, in Copley Square, is near the site of the April 15 bombings that killed three people and wounded more than 260 others.

“After those meetings, I will consider all of what we have heard, and come up with a final protocol,” Feinberg told The Associated Press.

The Boston Globe reported Friday that the families of those who died and survivors who lost more than one limb in the attack could be in line to receive more than $1 million from the fund, while people who lost a single limb could receive amounts approaching $1 million.

Feinberg said while estimates in the report were correct, he stressed that a final determination would not be made until after next week’s meetings, which are open to the general public as well as victims and their families.

The current schedule calls for claims to be filed May 15-June 15, and the first checks cut by June 30, he said.

Donations were planned to be accepted through Labor Day, but Feinberg said Friday there was a possibility The One Fund, which was formed by Gov. Deval Patrick and Boston Mayor Thomas Menino after the attack, could eventually assume a “permanent, ongoing role in providing communitywide assistance.”

Organizers of the May 30 concert at TD Garden, called “Boston Strong: An Evening of Support and Celebration,” said tickets for the event would go on sale Monday and that all net proceeds would go The One Fund.

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