Those looking at taking on the company are said to come from inside and outside the textiles industry and from locations around the world.
Accountancy firm KPMG said it was now setting a closing date for offers of 4pm on Wednesday, and is confident of selling Dawson as a going concern.
It is not yet known if management at Dawson are among those considering striking a deal.
Once a preferred bidder is selected it may still take several weeks before any transaction is completed but KPMG expects new owners to be in place before the end of September.
The announcement about the level of interest and the quick timescale for a sale raised union hopes that the Barrie Knitwear business, which employs 180 in Hawick, would survive unscathed.
Alex McLuckie, from the GMB Scotland union, said: "We are fairly hopeful a production base will be maintained in Hawick and there will be continuing employment for the people there."
Yesterday, Blair Nimmo, head of restructuring at KPMG and administrator of Dawson, said: "Following an extensive marketing campaign there has already been strong interest in Dawson International from potential buyers, with 35 separate enquiries having been made about the business.
"We have now sent detailed information packs to these interested parties. While encouraged by the level of interest we have already received, our priority remains finding a genuine and sustainable bid. As such we have set a closing date of August 29 for indicative offers.
"We remain confident in achieving a sale of the business as a going concern. However it is likely that it will take some time after the closing date to conclude and announce a deal.
"We are aware how critical the future of Dawson International is to employees, customers and suppliers and want to allow time to achieve the best possible deal for the company."
Dawson was forced into administration on August 15 after being unable to find a way to plug a £129 million deficit in its pension scheme, which has around 3320 members, although only 56 of those are active.
In July, Dawson failed to agree a rescue package for the scheme with the Pension Protection Fund (PPF), which provides compensation to members of eligible schemes where there is a qualifying insolvency.
It is thought Dawson proposed making a one-off payment to the PPF and awarding it a 33% stake in the group in the hope this would allow it to trade through its difficulties, but the fund refused. Around 1900 former Dawson employees now working for other companies are facing cuts of at least 10% in their pensions.
The sale of the Dawson business is unlikely to have any material impact on the pension fund.
Dawson's United States arm Dawson Forte, which has its main office in Boston and a showroom in New York, continues to trade and has not been affected by the administration.
Dawson was once owner of names like Pringle and Ballantyne and can trace its origins back 140 years.
