Friends Claire Henderson and Emma Neilson are hoping to secure a deal to supply House of Fraser's 62 stores around the UK with the Dick Winters range.
The talks with Frasers' buyers in London come after the brand, whose products are marketed under such cheeky names as Clever Dick, Sporty Dick and Dicks of Distinction, made a cameo appearance in the Glasgow store.
The duo presented the range as part of House of Fraser Presents, a two-week showcase of new products organised by general manager Paul Mitchell.
It came after the two literally door-stepped Mr Mitchell and asked for an audience, after dropping in a product sample for his attention the day before.
The product, which appeared in store over the Valentine's period, retailed for £28 per pair, and proved to be a hit with a broad range of consumers.
A listing with the prestigious chain would cap a flying start for the business, which Ms Henderson and Ms Neilson set up a year ago after becoming friends through dropping off and picking up their children from school.
They hatched the idea for Dick Winters after discovering they shared the frustration of being unable to find products made in the UK that met their expectations on quality and ethics when shopping for their partners.
They hope to provide an antidote to "abs fatigue", a reference to the way designer underwear for men was marketed using muscle-bound models.
Beyond the business concept, the two entrepreneurs were looking for a new challenge and to return to the workplace after focusing on bringing up their young children.
Coincidentally, they discovered through their own friendship that their husbands shared a chemistry classroom in their school days.
The impetus for the business was also informed by research that suggests more than half of women buy underwear for their partners, and that six in 10 receive it as a gift.
Ms Henderson said: "We did lots and lots of research and came across a gap in the market - there didn't appear to be any men's luxury boxer shorts that were available on the high street that were made in the UK.
"We felt that we would be our customer, in that if there was something made in the UK that was a higher quality, we would buy it."
The business owners carried out extensive research prior to the launch, which involved asking focus groups of men and women what they like and dislike about boxer shorts, from the quality of the material to comfort and style.
Ms Henderson noted: "It was deconstructing it and basically saying, 'What makes the perfect pair of boxer shorts?' It was interesting listening to both groups because both are the consumer."
The entrepreneurs' preperation reflected the experience they had built up earlier in the careers.
Ms Neilson gained a broad base of skills by working in the family fish merchant business in Ayr, while Ms Henderson's career was seen her variously work as an interior designer and for the Scottish University for Industry.
Ms Neilson said: "I was selling to chefs and fryers one minute, and then jumping in a van and delivering fish and chasing money. Now going to my own business I realise what a great grounding it was."
The two were open to ideas and benefited from a spell on the Entrepreneurial Spark (E-Spark) incubation programme in Glasgow, which they joined in August.
Although they had already trademarked their brand and lined up a manufacturer prior to joining E-Spark, it helped by putting them in touch with helpful mentors. They also benefited by experiences with the people behind other start-ups.
Ms Henderson said being on E-Spark "normalised the fear" of starting a business, stating: "You are all out of your comfort zone.
"If there were mistakes that any of us made, or quicker ways of doing things, the benefit is that kind of knowledge sharing."
The Dick Winters range, which is already sold by an independent menswear store in Bearsden, near Glasgow, is currently being manufactured in England.
However, the owners are hopeful of starting production in a factory in Scotland soon.
They are also in the process of securing a unit to serve as a stock-holding facility in Glasgow, and hope to take on staff in roles ranging from online order fulfilment to sales and administration as the business grows.
As for the rationale behind the cheeky name, Ms Henderson said: "We wanted to look at what made us unique, and it is our quirky sense of humour, hence the Dicks of Distinction.
"We've got Sporty Dick for the guy who is into sport, but again the product is a very sporty fabric - you could quite easily go out running and skiing in them.
"We've got Old Dick, for significant birthdays, retirements; Clever Dick, for new dads, exam success, promotions. Aside from that they lend themselves to gift buying opportunities, but actually also for the individual it's a bit of humour for them - they can buy Lucky Night for their Saturday nights, Rock Dick for festivals."