TL;DR - Almost certainly not.
20th Century Fox are the current owners of the live action rights to the character of Wolverine, and have been since the early 90's when Marvel sold the live action rights to many of their characters. Further details about the nature of these contracts (specifically the concept of the rights reverting to Marvel Studios after a set amount of years) can be found in my answer here.
The upcoming Wolverine film (spin off from the main X-Men films and semi-sequel to X-Men Origins: Wolverine and The Wolverine) sounds as if it may use elements from the Old Man Logan storyline according to Hugh Jackman himself.
Speaking to a massive crowd in Hall H, Jackman said, “I’ve got three words for you guys: Old Man Logan.” The actor then added, “Take from that what you will.”
Quote courtesy of this ScreenRant article
As such, Old Man Logan is almost certainly classed as a version Wolverine from a legal perspective.
However, a recent deal between Marvel Studios and Sony Pictures (who own the live action rights to Spider-Man) has resulted in Spider-Man becoming a part of the Marvel Cinematic Universe in the upcoming Captain America: Civil War. Although the rights to the character still belong to Sony Pictures, the deal means that each studio can use characters and events from the others films in order to bolster their own films, so we can expect to see Spider-Man appear in future Marvel Cinematic Universe films, and the upcoming Spider-Man films to reference events from the Marvel Cinematic Universe and potentially use character from other films.
Despite this deal giving many people hope that we will soon see mutants in the Marvel Cinematic Universe, the fact that the X-Men movies have been more successful for 20th Century Fox than the recent Spider-Man movies have been for Sony Pictures means that the studio is under no financial pressure to make a deal. As such, as far as we as the general public are aware, there are no plans (despite it being possible) to introduce mutants into the Marvel Cinematic Universe, which has resulted in Marvel Studios attempting to use their Inhumans property as a mutant substitute for now.
It is also worth mentioning that the inclusion of Quicksilver and the Scarlet Witch in Avengers: Age of Ultron is not indicative of a 20th Century Fox/ Marvel Studios deal - these characters are mutants first and Avengers second, meaning that both studios own the rights to the character and can use them as they see fit as long as Marvel Studios don't reference mutants and 20th Century Fox don't mention The Avengers. This has led to the potentially confusing situation where two different versions of the same character have appeared in two unrelated films (Quicksilver in both Avengers: Age of Ultron and X-Men: Days of Future Past).