UPDATE 1-Rhoen's biggest shareholders sell to Fresenius

* Eugen Muench and his wife tender 12.5 percent stake

* Swedish pension firm Alecta sells 9.1 percent stake

* Sales put Fresenius well on way to take over Rhoen (Adds Alecta statement, background)

FRANKFURT, June 18 (Reuters) - Rhoen-Klinikum AG's two biggest shareholders are selling their holdings to Fresenius SE, putting the German healthcare group well on its way to taking over the hospital operator.

Fresenius in April unveiled a plan to buy Rhoen for 3.1 billion euros ($3.9 billion), which would make it by far the largest private-sector operator of German hospitals.

Fresenius, which controls dialysis specialist Fresenius Medical Care, said on Monday that Rhoen founder and Chairman Eugen Muench and his wife have tendered their 12.5 percent stake to Fresenius, as expected.

Separately, Swedish pension firm Alecta said it had decided to sell its 9.1 percent stake in Rhoen to Fresenius as well.

"We are confident that Fresenius-Helios and Rhoen-Klinikum together will reach the targeted structure and results set forth for this transaction," Alecta portfolio manager Leif Toernvall said in a statement.

Rhoen and Fresenius's Helios division compete with German private-sector peers Asklepios and smaller Sana Kliniken, both unlisted.

The proposed Rhoen takeover represents a rare growth opportunity for Fresenius. German hospital privatisation has been slow due to the need for millions of euros of investment at each potential target and the prospect of laborious restructuring efforts following any takeover.

Rhoen is one of Germany's largest private hospital operators, with 53 hospitals, 39 health-care centers and 2011 sales of 2.6 billion euros.

The acceptance period for Fresenius' 22.50 euro-per-share offer ends on June 27. If the acceptance threshold of 90 percent is reached by then that period will be extended by two weeks.

Other shareholders of Rhoen include SEB with about 7.6 percent, or 12.4 percent including options, and Swiss lender UBS with 5.2 percent.

U.S. investor John Paulson, who holds 3.6 percent of Rhoen, said last week he will tender his 3.6 percent stake in the company. ($1 = 0.7921 euros) (Reporting by Maria Sheahan; Editing by Hans-Juergen Peters)