Mr. Bennet in Chapter 57 describes a letter from Mr. Collins this way: "He begins with congratulations on the approaching nuptials of my eldest daughter, of which, it seems, he has been told by some of the good-natured, gossiping Lucases. I shall not sport with your impatience by reading what he says on that point. What relates to yourself is as follows: 'Having thus offered you the sincere congratulations of Mrs. Collins and myself on this happy event, let me now add a short hint on the subject of another; of which we have been advertised by the same authority. Your daughter Elizabeth, it is presumed, will not long bear the name of Bennet after her elder sister has resigned it, and the chosen partner of her fate may be reasonably looked up to as one of the most illustrious personages in this land.'"

The exact mechanism for getting the rumor started is not clear, but we have other examples in P&P of rumors that blew up out of nothing (like Bingley's five sisters), so I don't think it was based on anyone's actual observation of Lizzy and Darcy's behavior but rather from someone talking about Jane and Bingey's engagement going to talking about what a nice-looking couple the bride's sister and the groom's best friend would make to eventually an engagement. As pointed out below, Sir William has pushed this in the past.