Elliott Management made an all-cash deal for Athenahealth, with a bid of $160 a share, the activist firm said in a letter Monday.

The proposal is a 27 percent premium to Athenahealth's share price as of Friday. The shares jumped nearly 24 percent in trading on Monday after CNBC's David Faber reported that Elliott was preparing to make the bid, citing sources familiar with the matter.

Paul Singer's Elliott, which took a 9.2 percent stake in the health-care technology company about a year ago, said Monday its stake was now 8.9 percent. Athenahealth is led by CEO Jonathan Bush, the outspoken and controversial cousin of President George W. Bush and brother of Billy Bush.

In the letter to Athenahealth's management, which Elliott released publicly on Monday, the activist firm said it has had constructive discussions with management, "and we greatly appreciate how much time the management team, led by Jonathan, and the Board have invested in evaluating our perspectives as well as the perspectives of our fellow shareholders."

In a statement, Athenahealth confirmed it received an unsolicited bid from Elliott and said its board will "carefully review" it. It hired Lazard as financial advisor and Weil, Gotshal & Manges as legal advisor.

Elliott's letter also says Athenahealth's management has "revealed an unwillingness" to pursue alternative strategies for realizing value. Elliott approached the company last fall about a take-private transaction involving Elliott or others and the company "refused to engage."