, which has 21 stores across England and Scotland, confirmed it had been approached at the end of the May about a takeover, but it was initially thought a private equity group would acquire the Scottish-based chain.
Tesco has since confirmed its bid, which is understood to be around £156m, has been accepted by Dobbies. The supermarket already had a 22.6% stake in the business.
According to Tesco, Dobbies' strong brand name and respect within local communities were part of its reasons for entering into the gardening sector with the acquisition.
Sir Terry Leahy, chief executive officer at Tesco, said: "The deal is an important part of our strategy to provide customers with greater access to affordable energy saving and environmental products.
"Garden centres are ideally placed to support this, because for many people, gardening is the way they express their desire to be green."
Dobbies, which is primarily based in and around Glasgow and Edinburgh, also has outlets in Shrewsbury, Milton Keynes, Cirencester and Reading. The business specialises in gardening ware for homes, including outdoor furniture, foliage and accessories.
Tesco's bid represents a share price value of around 1,500p a share for Dobbies, which reported a 46% rise in pre-tax profits in April to £2.5m.