The general consensus is that he may have been engaged to Jessie Thain, though I have never seen a primary source proof that would cinch this. Some editions of Alexander Smellie's biography of R. M. McCheyne (a burning light) have as an appendix, Extracts from the diary of Jessie Thain.
His note prompted me to Google for her name and I came across this gem, from which I include a brief quotation below.
In his interesting book, “Robert Murray McCheyne,” Dr. Smellie mentions the fact that McCheyne was twice engaged to be married. The excellent young lady whom he first honoured with an offer of his life and love was discouraged by her relatives who thought they saw in McCheyne’s frail body a foreshadowing of an early death. The other young lady was the writer of this Diary. Certainly her references to McCheyne’s death in her Diary are strongly suggestive of a relationship nearer than that of a pastor to his flock. Spiritual affection for those in whom we see the image of the Lord is an ennobling grace which may not always be free from pain; but the love of Jessie Thain for Robert McCheyne, while holy and intensely spiritual, appears also to have the additional marks of the fond and reciprocating attractions of nature. One cannot but admire the delicacy and refinement with which she touches on this subject.The online page from which this quotation is taken is entitled, Jessie Thain (The Friend of Robert Murray McCheyne) - Edited by Rev. Murdoch Campbell, M.A. The editor's introduction is dated Ross-shire, September, 1955.
Miss Thain was born 31st July, 1820, and was thus seven years younger than the one she loved. Her diary covers the period 31st December, 1843 to 28th November, 1847. M'Cheyne died on 25th March 1843, aged only 29, and there is a diary entry for the same date in both 1844 and 1845. It makes poignant reading.
NB. A 2003 paperback edition from Shiloh Publications is SOLD OUT. There were earlier editions in 1961 and 1967, as well as the originally edited one of 1955. These may sometimes appear in used book stores, whether online or in retail shops.