Neither Pret nor Eat have commented on the rumors, but according to the Financial Times, the 'details were being ironed out'. The paper added that there are other potential bidders for the chain.
Pret has around 400 outlets in the UK - four of which are vegan and veggie-only stores - called Veggie Pret. Three are based in London and one in Manchester.
Now according to the Evening Standard, Pret is currently looking at buying either the majority of Eat stores, or the business in its entirety. The Telegraph says: "[Pret has] a view to converting most of [Eat's] 90-odd stores to the Veggie Pret brand."
Pret a Manger opened its first Veggie Pret in Summer 2016. It was supposed to be a month-long experiment, with the chain expecting to see losses.
Writing a blog post, Pret CEO Clive Schlee, said: "What is it about the shop that has struck such a chord? Clearly, the move towards a plant-based diet is gathering momentum, especially among millennials.
"Vegetarians and vegans are an articulate and social media-savvy group. Many of you have been advocating eating less meat for years and have been thrilled to see a major UK business take up the cause.
"We have had over 20,000 customer comments in the past twelve weeks and many of you have offered heartfelt thanks to Pret for making a bold move. Even Sir Paul McCartney emailed."