His assertion comes after a number of organisations in the sector signed a joint letter expressing concern about the negotiations, claiming they have been rushed and could set a bad precedent.
Mr Jack said he has been involved in discussions around the post-Brexit trade deal, which he said are nearing completion.
He said safeguards against the UK market being “swamped” with imports will be put in place.
Speaking to BBC Scotland’s Sunday Show, he said: “We’re very close and it’s going to be a very good deal for the UK.
“I think it will also take us into the Trans-Pacific Partnership, it’s gateway deal into that, which is a huge, huge opportunity for Scotland’s food and drink industry.”
He said an assessment from the Department for International Trade had shown Scotland would “benefit more than any other part of the UK” from the deal.
Mr Jack said the UK Government would not discuss the details of the negotiation in public, but added: “We won’t be taking chlorinated chicken in any trade deal we do, that’s illegal in the UK.
“We won’t be taking hormone-induced beef, that’s illegal. We will have safeguards in around the amount of product coming, so we won’t see the market swamped.”
UK international trade secretary Liz Truss has previously suggested the 5 per cent whisky tariff could be scrapped under the deal.
Scottish food producers had sent a letter warning Ms Truss about the way trade deals are being negotiated last week.
Earlier this week, Scottish rural affairs secretary Mairi Gougeon wrote to Ms Truss to seek an urgent meeting on the deal.
She said: “In their letter, industry representatives have raised concerns at the lack of consultation and have asked that you publish an assessment of the cumulative impact of FTAs with Australia, Canada, New Zealand and the United States.
“The Scottish Government would echo their ask.
“It is also clear that industry representatives share the same concerns that I raised in my earlier correspondence on animal welfare, environmental and food safety standards.”
UK foreign secretary Dominic Raab