In a letter written in Tamil, Modi wished the chief minister good health and happiness "to serve the people," an official release here said.
The chief minister had turned 63 on May 12.
"I am happy to extend my hearty greetings to you on your birthday.. Let the Almighty bless you with good health and happiness to serve the people," the release quoted Modi as saying.
The chief minister, in return, expressed delight over receiving birthday greetings from Modi in Tamil, it said.
"It was very gracious of you to have remembered my birthday and conveyed your greetings to me," the release quoted the chief minister as saying.