The Justice Department and Citigroup are close to a multibillion-dollar deal to settle allegations the bank sold shoddy mortgages in the run-up to the financial crisis, according to people familiar with the matter.

The two sides, which had been far apart just weeks ago, are now ironing out details of an agreement that would have the bank pay more than $4 billion to avert a federal lawsuit over the precrisis mortgages, these people said. A settlement could be announced as early as next week, according to people familiar with the matter.