The Greek term there at John 11 (both verses) is “in-thunder”. (Which is one of the more awesome sounding words in Greek, too: embrimaomai! Or as we might say here in the South, “Im Brim MahOhMy!” )
It absolutely does not have the connotation of praying (not in itself anyway), and even “muttering” is too weak and doesn’t quite get the emotional thrust correct. It’s a rumbling, threatening GROWL!–the sort of thing someone does when he’s ticced off.
A more accurate translation, including in grammar, from the Greek might be: “Then Jesus, as He saw her wailing, and the Jews wailing as they came toward her, growls in His breath [or ‘under His breath’ as we would say in English], shaking Himself.” And, “Jesus, deeply growling in Himself, is coming toward the tomb.” (Edited to add: This is how I translated it for my King of Stories harmonization project several years ago, by the way.)
It’s the same term used for Jesus when strictly warning two different men he had just healed not to go blabbing about this to the public right before throwing them out of the house (which they then go boast to the public about) at Matt 9:30 and Mark 1:43. By comparison, it’s also the term used by the disciples embrimaomai-ing against Mary for bringing the attar at Mark 14:5. (I think those are the only times it occurs in the NT.)
This use of the term has long been recognized, and one of the popular topics for interpreting and applying GosJohn in this chapter, throughout all Christian history, has been: what (and/or who) is Jesus angry about here, and why? (The text doesn’t say, so there has been a lot of debate and discussion. I strongly suspect more than one answer is correct, both concerning the target of His anger and His rationale for it.)