Earlier this year I expressed the opinion that they are probably not in Bethany and The Mount of Olives. But then I considered that they could be briefly in The Sisters of Jesus. Well here I want to consider how plausible this is independent of the greater points of either of those posts.
First of all, if you're thinking "the whole point of calling them Magdalene and Of Bethany is to distinguish between different women with the same name", congratulations, you just demonstrated some casual Biblical Illiteracy. The Text of the Gospels never actually uses "Of Bethany" as an epithet for anyone, we call Mary of Bethany that because we encounter her there, and "John" 11 & 12 implies that's where she and her siblings live. But no title is ever used of Mary of Bethany in any scene viewed as being about her, she's just Mary if she's named at all.
Tradition isn't very helpful here. Eastern Orthodox tradition distinguished them. Western/Catholic tradition often merges them but also throws in women I strongly view as separate, in my view neither Mary can be Identified with the accused Adulteress of John 8, and I'm likewise skeptical of doing so with the woman from Luke 7. And most traditions have the Bethany siblings traveling to other regions like the Twelve did while I believe their remains can be found in the Dominus Flevit Church on the Mount of Olives which means they probably never lest Judea, but it's not impossible.
Luke and "John" are the primary Gospels that need to be investigated here. Matthew and Mark's account of the scene from "John" 12 doesn't mention the woman by name, and in every Gospel but Luke the epithet Magdalene isn't used prior to The Cross.
The Gospel According to The Beloved Disciple (commonly called "John") can definitely be argued to make the most sense if the Sister of Lazarus and Martha is also the Mary called Magdalene. I have come to support the theory that Lazarus is the Beloved Disciple (though maybe sometimes that designation can be expanded to include his Sisters, but it can't include any of the Twelve), which makes it seem natural to conclude that one of the Marys in Chapter 19 Verses 25-27 is his sister. Or even more so if Mary rather then Lazarus is the one being called the Disciple whom He Loved there. And then in chapter 20 when the word "other" is added to a verse with Mary Magdalene implying two people in that verse are a "disciple whom Jesus loved".
Luke is basically the source of every argument against them being the same woman. Well I have some thoughts on that.
First of all, we proponents of Luke's reliability as a Historical document point out how he claimed to have gathered his information by interviewing multiple Eye Witnesses. So it's possible he himself may not have always known if some of the stories about characters with the same name were really the same person or not. Luke uses the title Magdalene only in verses that also mention Joanna, so it could be Joanna called her that but his source for the story about Martha & Mary in chapter 10 did not. And this could also explain the same characters being named in some scenes but not in others.
Luke 8 associates Mary Magdalene with Jesus Galilean ministry, while "John" 11-12 is taken as making it seem like the Bethany siblings were not involved in the Galilean ministry. But the Bethany siblings seem to be pretty well off financially given how the Ointments Mary used were so expensive. So maybe this house in Bethany wasn't their only residence but one they used mainly around the time of the Pilgrimage Festivals.
Bethany isn't mentioned in Luke 10, indeed just reading it in the context of Luke's narrative without bringing in assumptions from elsewhere it seems like this is in Galilee as much as everything around it was. The Synoptic narrative is uninterested in the Judean ministry prior to them arriving there for Passover.
Analyzing Luke as a stand alone document, the only real argument against Magdalene and the sister of Martha being the same Mary is a face value assumption that Luke 10 is painting Martha's sister as awfully passive compared to the active Disciples introduced in Luke 8. But that misses the actual point of what was going on in Luke 10.
Luke 8:1-3 is clear that there were more women then just the three who are named here, likewise with 24:10 and the other Synoptic accounts of the Myrrbearers. Maybe what goes on in Luke 10 explains why Mary but not Martha would be mentioned by name in those passages.
Some people do think the Penitent Woman in the house of Simon in Luke 7 is another account of the Anointing incident from "John" 12, Mark 14 and Matthew 26. I think there are too many important differences for them to be the same event, but the idea that they involve the same Woman isn't impossible. Well the argument for that woman being Mary Magdalene is that she's the first woman named in Luke 8 which immediately follows that story.
So looking into all the available information, it is certainly possible they are the same, but by no means proven beyond any doubt.