MSP John Lamont called for the move in a question to the Scottish Parliament.
Business Minister Fergus Ewing said it was something they would consider in the wake of the firm entering administration.
He added that he would visit Hawick on Wednesday to see first hand what could be done to help staff affected.
The historic Borders firm went into administration last week with the immediate loss of 123 jobs and 56 more under threat.
Mr Lamont asked what the Scottish government was able to do about the situation.
Mr Ewing said he was "deeply concerned" by the job losses and said he would be speaking to administrators when he visited the town.
He said they were "exploring every possible avenue of support" for the company and were looking to assist efforts to find a buyer.
The minister added that a redundancy support event and jobs fair would take place in the town on Friday.
Mr Lamont said the "proportional effect" of at least 120 job losses in Hawick was great and called for the task force to be created.
"We will consider that further," said Mr Ewing. "We are open-minded as to whether a task force is required.
"I think it is prudent first to take a little while to assess the situation, working closely with the administrators as we always do and the workforce representatives and the local authority."
To catch up on full details of the debate you can visit Holyrood Live.
The Scottish government has said it will consider setting up a textiles task force following more than 100 job losses at Hawick Knitwear.