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I know that it's correct if you say "Who am I speaking with?". What if I add the phrase "May I know" prior to "Who am I speaking with?"? should it be "MAY I KNOW WHO AM I SPEAKING WITH?" OR "MAY I KNOW WHO I AM SPEAKING WITH?"
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I agree with much of what Avangi says, but may I start from the beginning?

These are the generally accepted options, at two levels of formality:

Who am I speaking with? -- less formal, very common

With whom am I speaking? -- very formal, but receptionists are sometimes enjoined to use it.

It is generally considered 'incorrect' or at least badly mixed register to insist on the object pronoun 'whom' when it is not preceded by the preposition: 'Whom am I speaking with' is an attempt to be formal, but the terminal preposition in and of itself sets the sentence as informal.

Adding 'May I ask' ('May I know' is not a native collocation, to my knowledge) changes the sentence structure by becoming the question clause, and the subordinate clause (now a complement) reverts to statement word order:

May I ask who I am speaking with?-- less formal, very common

May I ask with whom I am speaking?-- very formal

That said, you will indeed often hear – and the punctuation should make the reason clear:

May I ask – Who am I speaking with?
May I ask – With whom am I speaking?
Comments  
I'm puzzled. I realize that "Who am I speaking with?" is acceptable, but I don't think it's grammatical.
Would you say, "I was speaking with he yesterday." ?

The preposition "with" takes the objective case." "Whom did you come with?"
"Did you come with him?" Not, "Did you come with he?"

(It's okay to ask "Who did you come with?")

Adding "may I know/ask" doesn't change anything.

"Who is this?" and "May I know who this is?" are both grammatically correct.
Likewise, "Who is calling?" and "May I know who is speaking?"
In the last two cases, "who" is the subject of a clause.
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 Mister Micawber's reply was promoted to an answer.

may i know to whom i am speaking?(correct)

may i know to whom am i speaking?(incorrect)

before getting to that i make it clear of this basic rule

"wh" question+helping verb+subject+main verb {always}

what are you doing

where do you play badminton

but in this sentence{may i know to whom i am speaking?}

whom act as "relative" pronoun and not as "questioning word"

to be more precise

after verbs of thinking:

know - understand - suppose - remember - forget - wonder'

we always consider who or whom as a relative pronoun so we must follow the general rule {sub+helping verb+main verb}

eg:i know who i am

i wonder what he is doing

look these sentence it is not a question because who act as relative pronoun not question word

may i know to whom i am speaking?(correct)

THIS VERBS OF THINKING ALWAYS TAKE THIS FORMAT

i am not expert in English I am also a learner just thought of helping which i know sorry if my language is pretty weak Emotion: smile

desk park 268i am not expert in English I am also a learner just thought of helping which i know sorry if my language is pretty weak

You could at least make yourself look as if your language were a little stronger if you would just use capital letters correctly. The word I is always I, not i. Begin your sentences with a capital letter. Don't use capitals all the way through a sentence. Also, don't forget to put a period at the end of your sentences (or a question mark if necessary).

If you only do these few things, your posts will appear much more professional, and even if you're not an expert, people will think you have something important to say and they will enjoy reading it more. Emotion: smile

CJ

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Never end a sentence with a pronoun.

Are you talking to me? Emotion: wink