In Burnside's own testimony on Dec. 19, '62, before the Joint Congressional Committee, he stated the following:
…'On the 7th or 8th of November I received an order from the President of the United States directing- me to take command of the army of the Potomac, and also a copy of the order relieving General McClellan from that command. This order was conveyed to me by General Buckingham, who was attached to the War Department. After getting over my surprise, the shock, &c., I told General Buckingham that it was a matter that required very serious thought; that I did not want the command; that it had been offered to me twice before, and I did not feel that I could take it. I consulted with two of my staff officers in regard to it for, I should think, an hour and a half. They urged upon me that I had no right, as a soldier, to disobey the order, and that I had already expressed to the government my unwillingness to take the command. I told them what my views were with reference to my ability to exercise such a command, which views were those I had always unreservedly expressed — that I was not competent to command such a large army as this. I had said the same over and over again to the President and Sec retary of War, and also that if matters could be satisfactorily arranged with General McClellan, I thought he could command the army of the Potomac better than any other general in it. But they had studied the subject more than I had, and knew more about their objections to General McClellan than I did.
There had been some conversation in regard to the removal of General McClellan when he was bringing away his army from before Richmond. The first of these conversations with the President and Secretary of War occurred at that time. And then, after General McClellan had got back to Washington, and before the commencement of the Maryland campaign, there was another conversation of the same kind. And on both of those occasions I expressed to the President the opinion that I did not think there was any one who could do as much with that army as General McClellan could, if matters could be so arranged as to remove their objections to him….'
(Report of the Joint Committee on the Conduct of the War, 1863, Part I, at page 650)
According to Burnside's own words above, Lincoln in two separate conversations offered him the army command, firstly following the 'Seven Days Battles' (June 25 – July 1, '62) and then again, prior to the start of the 'Maryland Campaign' (Sept. 4 – 20, '62). (In 'To the Gates of Richmond' at page 351, Stephen Sears mentions that this first conversation occurred several days before Halleck's appointment on July 23, as general-in-chief of the Union armies).
No exact dates given, though.