WASHINGTON — U.S. banks are enjoying their best profits in six years and are lending a bit more freely. The gradual improvement suggests that the industry will sustain its healing from the worst financial crisis in decades and help strengthen the economy.

The industry earned $37.6 billion from July through September — a 6.6 percent increase from its earnings in same quarter last year. For the first time since 2009, the stronger earnings were due mainly to higher revenue rather than to less money set aside by the banks to cover losses, data issued by the Federal Deposit Insurance Corp. showed Tuesday. And loans to consumers rose nearly 1 percent from the July-September period of 2011.

Some of the largest banks are cautioning, though, that their earnings are up mostly because they’ve sold less-profitable businesses, shed bad loans and trimmed jobs — not because of a more vibrant economy.