Dumbledore asked Snape during the first two weeks of July '96
In Harry Potter and the Half-Blood Prince, we learn that Dumbledore got a black hand he did not have by the end of Order of the Phoenix:
As [Dumbledore] replaced his wand in his pocket, Harry saw that his hand was blackened and shriveled; it looked as though his flesh had been burned away.
"Sir - what happened to your - ?"
Harry Potter and the Half-Blood Prince, ch.3 Will and Won't, p39, First American Edition
We also learn that good ol' Albus came to fetch Harry in the beginning of July :
It just seemed too good to be true that he was going to be rescued from the Dursleys after a mere fortnight of their company.
Harry Potter and the Half-Blood Prince, ch.3 Will and Won't, p36, First American Edition
This is confirmed by the narration stating that Harry
remained within the confines of the Burrow's garden over the next few weeks. [...] To Mrs Weasley's displeasure, Harry's sixteenth birthday celebrations were marred by grisly tidings [...]
Harry Potter and the Half-Blood Prince, ch.6 Draco's Detour, p78, First American Edition
Since Harry's birthday is the 31st of July, and the narration stating "few weeks", he must have arrived at least two weeks before that.
In Deathly Hallows, during the trip in Snape's memories, we learn that Dumbledore asked Snape to mercy-kill him right after Severus saved him from the curse the Ring of Gaunt carried.
His right hand dangled over the side, blackened and burned. Snape was muttering incantations, pointing his wand at the wrist of the hand [...].
Snape raised his eyebrows and his tone was sardonic as he asked, "Are you intending to let him kill you?"
"Certainly not. You must kill me."
Harry Potter and the Deathly Hallows, ch.33 The Prince's Tale, p481 - p483, First American Edition
Thus we can infer that the question came between the Ending Feast of Harry's fifth year and the end of the second week of his holidays.