The drilling rig was destined for a yard in Turkey to be decommissioned when a towline broke and it ran aground on Lewis.
It is now temporarily anchored in the island's Broad Bay where divers will inspect the structure for damage.
But councillor Angus Campbell has said it could remain longer term.
He has written to the rig's owner, Transocean, to consider breaking up and recycling parts of the structure at facilities near Stornoway on Lewis.
Hugh Shaw, the Secretary of State's representative for maritime salvage and intervention, who is overseeing the salvage operation, has said a decision on the final destination of the rig has still to be made.
In his letter to Transocean, Mr Campbell said: "We are pleased that you continue to engage with the local supply chain and, given the significant and positive economic impact to the islands, would now ask that serious consideration be given to the use of local facilities and engineering expertise in carrying out the necessary repairs.
"As you will be aware, there is an established oil and gas supply chain in the islands with considerable lay down area at the Arnish facility and good connections for skilled workers to move in and out as required.
"Secure additional storage for recyclable parts pending resale could be available at Creed Park on the outskirts of Stornoway.
"We would be very happy to work with Transocean, Smit Salvage and other relevant key stakeholders to ensure that direct and indirect benefits are, as much as possible, retained in the islands."

The victim, Jason Bryden, 38, was repeatedly stabbed during the attack at his home in Kilmarnock on 5 July 2015.
At the High Court in Glasgow, James Clark, 36, was jailed for a least 22 years after being convicted of murder.
His brother John, 38, was jailed for 10 years after being convicted of the lesser charge of culpable homicide.
Jailing both men, judge Bill Dunlop told the brothers that they had carried out "a merciless, relentless and savage attack".
He rejected any suggestion that Mr Bryden had been laying in wait for them armed with a samurai sword.
Judge Dunlop added: "You together planned to gain illegal access with at least one knife and nunchucks.
"Jason Bryden may have been a drug dealer but he was a son, a father and a friend. He was butchered by you."
After both men were jailed, Det Insp David Moran, who led the investigation, said: "The violent and terrible actions of John and James Clark caused the death of Jason Bryden.
"To be taken at the age of 38, in such horrifying circumstances, is a heartbreaking situation that the family has been forced to endure."
The court previously heard the brothers, from Kilmarnock, had decided to rob Mr Bryden of a gold bracelet which they had given to him in exchange for drugs.
The pair, who were high on drink and drugs, went to his home after masking their faces with scarves and arming themselves with a knife and martial arts nunchucks.
The sword used in the killing is believed to have belonged to the dead man.
Neighbours said they could see the Clarks attacking Mr Bryden as he sat on a sofa by the window and could hear him screaming.
Prosecutor Richard Goddard told the trial: "This was a sustained, frenzied attack on this man."
Afterwards, the scene of the killing was described as a "bloodbath".
In addition to the stab wounds, Mr Bryden suffered blunt force trauma during the assault.
He died in Kilmarnock's Crosshouse Hospital three days later.
Both brothers denied the killing.
In evidence, James Clark, who has 38 previous convictions including one for attempted murder, said: "I didn't mean it. It was an accident."
John Clark, who has 23 previous convictions, said he only got involved to help his brother whom he claimed was being attacked by Mr Bryden.
The leader of Western Isles local authority, Comhairle nan Eilean Siar, has said the rig Transocean Winner could be scrapped on the islands.
[NEXT_CONCEPT]
Two brothers have been jailed for killing a man with a samurai sword after a drugs-related robbery in East Ayrshire turned into a "bloodbath".