The Mongolian armies (according to Jack Weatherford in Ghengis Khan and the Making of the Modern World) would sometimes do as you suggest, giving towns the option to join his (and other Mongolian) army.
One of the most interesting facets of the Mongolian invasions is that they would sometimes pass a city by without bothering it and sometimes would attack without warning. A city targeted by the Mongolians would often be treated in a way that corresponds to how they react to the Mongols. For example, if a town offered resistance, they'd be treated without pity, but if a town welcomed the Mongols (or at least did not fight against them) they may be left to their own devices, as long as they acknowledged the Mongols superiority. It is this arbitrary attacking of towns that helped to create the sense of terror surrounding the Mongols. (my copy of Ghengis Khan and ... was stolen by a student (I teach APWH) so I do not have the complete information to cite.)

Obviously noticeable is my lack of flair, but the book on Genghis by British travel writer and historian John Man and his Mongolian sources from going there himself say that yes indeed the mongol army gave a lot of castles and cities a choice to surrender their supplies, army, and women for survival or to let the Mongols take it by force.

Now, the mongol army didn't necessarily give every single settlement they conquered this option, but when they didn't it was mainly due to the nature of negative diplomacy from their ruler. If someone upset the Kahn, he would have no mercy for the ruler or their citizens.

They tended to give the option to surrender or get slaughtered during the early reign and campaigns in modern day Chinese lands. When the mongol horde turned their gaze west, and had poor relations with Muslim and Christian rulers, they were not so keen on giving a choice to the settlements.

Something you may be interested in:

An extract from Simon of St Quentin's account of his visit to the Mongol Court (1248), recounting a letter from the Great Khan to his General:

Draft of the Chan's letter to Baiothnoy, which the Tartars call the Letter of God :

It is our will that this reaches the ears of all men in everyplace, both in provinces subject to us and in provinces that are in open rebellion. It is your duty Baionoy, to rouse and inform them that this is the edict of their living and immortal God...Whosoever shall gainsay you, let him be hunted down and his land devastated. I tell you this, that whoever does not hearken my decree shall be deaf, whosoever shall see my decree and not act it out shall be blind, and whosoever shall act in accordance with my judgement, knowing peace yet not observing it, shall be lame. Let this order of mine reach every man. Whosoever hears it and fails to observe it shall be destroyed and shall perish... And whosoever is intent on the advantage of his house and pursues it and is willing to serve us, let him be spared and shown great honour. But whosoever refuses to hearken, do as you will with them.

Source:

Narrative of Simon de Saint-Quentin, member of the mission of Friar Ascelin to the Mongol general Baiju on behalf of Pope Innocent IV (1248) from Vincent de Beauvais, Speculum Historiale (c1255), ed. by Johann Mentelin (Strassburg 1473) ; extracts edited by Jean Richard, Simon de Saint-Quentin. Histoire des Tartares (Paris 1965), pp. 94-117.