The rig, called Kulluck reconnected with the tug vessels Aiviq and Alert overnight Sunday, the Anchorage (Alaska) Daily News reported.
Towlines between the boats and the drill rig were lost Sunday afternoon due to forceful winds and high seas, the newspaper said.
A statement issued Monday morning by the unified command overseeing the situation said the Alert "was able to secure the 400-foot line that was previously the towline used by the Aiviq. The Alert successfully added tension to the line to test its ability to hold."
"Difficult weather conditions are anticipated to continue over the next several days. Unified command is evaluating all potential options to further secure the vessel until the weather clears," the statement said.
The rig's 18 member crew was evacuated Saturday as a safety precaution after the rig began drifting.
Sean Churchfield, Shell's incident commander and the company's operations manager for Alaska said Sunday that a full investigation into what caused the towline mishaps has been started.
"I don't really want to speculate as to the causes of the propulsion failure on the Aiviq," Churchfield said. "We are looking for the solutions and we will have a full investigation. At this stage, I don't have any firm information to pass onto you."
Swim Week Miami Beach 2014 [PHOTOS]