Kindred Healthcare Inc. offered to buy a 14.9 percent stake in Gentiva Health Services Inc. for $16 a share as the company seeks to become the largest shareholder in the provider of home health-care services.

Kindred won’t consider further increases in its proposed price unless Gentiva begins discussions and shows that the company is worth more, Louisville, Kentucky-based Kindred said today in a statement. Gentiva’s closing share price on July 11 was $15.82.

Gentiva turned down an offer in May of $14 a share, or about $533 million, from Kindred, a provider of long-term medical care.

Citigroup Inc. is acting as financial adviser to Kindred.