They don't have a lot of interest or ability to do so, which is why they didn't OTL.
I'll have a go, though. Let's say Louis survives Mohacs and becomes a puppet ruler for Suleiman the Magnificent, who then has more sense (and luck) later in the taking of Vienna which is tentatively held. Following victories in Malta, Southern Italy and the Indian Ocean and after increasing his influence in Northern Africa the Ottoman Empire is overextended but just shy of comfortably so. The areas conquered can be held at a cost deemed low enough to accept and no further expansion is considered either necessary or wise.
Suleiman is old and begins to focus on domestic concerns, securing the borders, improving the infrastructure, maintaining the economy, etc. In doing all of these he is advised that one thing is perhaps the most vital: a canal or system of canals linking the Red Sea to the Mediterranean.
The concept will begin with a Red Sea to Nile canal system in the short term to ease the burden of the construction of a much longer term Suez Canal. The massive project begins and Suleiman dies.
His son Sehzade Mustafa comes to power following his father's death and, with the exception of some minor wars with the Safavids is left to continue to secure the empire and promote the economy. The mega project continues apace but with the Mediterranean essentially an Ottoman lake and the Indian Ocean free of Portuguese he decides to get him some of that gold over in the new world, partially to retard European settlement, partially to contribute to the economy, partially to increase the ottoman presence in the Pacific, and partially to find a place for the mass of undesirables the ottomans have collected from their new territories in Europe and Africa.
While by the mid to late 16th century much of the New World has been claimed by Euros, there's still plenty they either don't know about or have only nominally claimed with no power projection to back those claims up. A few trading port towns in the Caribbean and South America are likely with a better Ottoman hold on the Maghreb but the real region not yet secured by the Spaniards and only barely known to the Portuguese and French would be the American West and the Californias in particular.
Let's assume that by the end of Sehzade's reign the Suez Canal and of course its earlier phase connecting the Nile to the Red Sea canals are all completed. With Caliphornia (see what I did there?) essentially an Ottoman penal colony for some Turks and Arabs but mostly Balkan, Italian, Hungarian, Berber, and African dissidents as well as religious detractors, fur traders, youngest son Bedouin, adventurers, and other normal settlers it won't be long before gold is discovered and the Ottomans begin sending hordes of troops to defend the prospectors that flood in from the OE via the canal.
So now you've got the American West an Ottoman colony. The powder keg that represents is tantalizing but I'll stop.
None of the above is terribly likely, but none of it is impossible. Some of it I'm working on using for a TL I'm developing, but I won't say how much and I won't give away the POD that allows for Suleiman's good fortune...
Anyway, hope that gets the discussion going for you.