When Elizabeth first spoke to William Maitland of Lethington, the Scottish ambassador, about the potential marriage match, he laughed it off and then asked why Elizabeth herself did not marry Dudley; that way she could leave both her husband and kingdom to Mary, Queen of Scots! Elizabeth's chief advisor, William Cecil, supported the plan. After all, it killed two birds with one stone: it got rid of the troublesome Dudley and it formed an alliance with Scotland. Cecil wrote to Maitland in praise of Dudley but Maitland did not pass on the proposal to his queen because Dudley was not even a peer. He was a nobody. This was something that Elizabeth said that she would rectify by giving Dudley an earldom. However, although Elizabeth's ambassador to Scotland, Thomas Randolph, was ordered to keep on urging the Scottish queen to marry Dudley, neither Mary, Queen of Scots or Robert Dudley were keen on the idea.
In the spring of 1564, when Randolph spoke to Mary of the idea, she was flabbergasted. She had been married to the King of France, and here was Elizabeth trying to marry her off to a nobody. She did not say an outright "no", and demanded that if she were to marry Dudley then she should be named Elizabeth’s heir, but this was something that Elizabeth would never consider.
The marriage never took place, and it is hard to know how serious Elizabeth was in her offer of Dudley, her favourite, to her nemesis, Mary, Queen of Scots.
(Extract taken from Claire's book On This Day in Tudor History)