Ae Fond Kiss, one of Burns' best-known works, was written for the busty young Edinburgh beauty. Their love for each other was spelled out in their lavish exchange of letters.
The affair was at its peak while Burns was spending much of his time in a Glasgow hostelry, the Black Bull Inn and he is known to have written at least one of his letters from there.
The Black Bull had 32 bedrooms and was built for the Highland Society in 1758.
It stood in Argyle Street between the junctions of Glassford Street and Virginia Street and was one of the most fashionable hostelries of its time.
At the corner of Virginia Street, on the wall of what is now a Marks & Spencer store, is a plaque commemorating the poet's visits there.
It says: Robert Burns lodged here when this building was the Black Bull Inn. He visited Glasgow June 1787, February and March 1788."
The tablet was placed there by the Scottish Burns Club.
In fact Burns was a more regular visitor to Glasgow than the plaque indicates. There are at least five recorded visits to the city - and probably more.
He is known to have spent a rollicking good night in the Black Bull with his friend from Irvine, Captain Richard Brown, the man credited with persuading Burns to get his work published.
The Black Bull was built by John Glassford, the well-known Glasgow merchant after whom Glassford Street is named.
In the late 1700s the coach between Edinburgh and Glasgow picked up and dropped off passengers there.
Agnes McLehose, although she lived in Edinburgh, was originally from Glasgow, the daughter of surgeon Andrew Craig.
Her husband James McLehose had deserted her but sadly the love affair her Clarinda enjoyed with Burns' Sylvander never flourished.
Even though she was alone, the fact she was still legally married caused her to spurn the poet's advances. Just before her death she wrote: "I parted with Burns in the year 1791, never more to meet in this world, may we meet in heaven."
Nowadays the plaque reminding us of Burns and his visits to Glasgow is prominent and easily seen by anyone walking along that part of Argyle Street.