Pelephone, a unit of Israel's largest telecoms group, Bezeq Israel Telecom, has nearly three million subscribers, many of whom were prevented from receiving or making calls when the network went down late on Sunday.
"Working hard through the night, we managed to locate and neutralize one of the components that malfunctioned," Pelephone CEO Gil Sharon told Israel Radio, adding that engineers from manufacturers Ericsson and HP assisted.
Sharon would not say how many customers were affected, but Israeli media said it was "millions."
He told reporters that customers could choose the compensation - 60 minutes of free calls within Israel or to the United States and Europe, or 500 MB of Internet surfing. He did not say how much it would cost the company.
Bezeq shares fell 1.4% in a flat Tel Aviv market on Monday.
Sharon ruled out Israeli media speculation that a cyber attack had brought the network down.
Sharon said it was too early to determine whether subscribers would be compensated and that the company was investigating the cause of the problem.
Bezeq's mobile phone unit has suffered intense competition from the entry of six new operators in 2012, which sparked a price war. Bezeq net profit fell 38% in the third quarter.
The company also faces increased competition in the landline phone market.