Let me know when you are free to discuss VS Let me know your convenient time to discuss which is right and polite

Question

When writing an email to a person to get a time to discuss some thing, what would the correct and polite way of writing among the below two. If both are wrong then what would be the right thing to say.

1. Let me know when you are free in this week to discuss

2. Let me know your convenient time in this week to discuss

Answer 1

The choice of wording depends on a few things, including:

1. Intent: Are you trying to nudge for a meeting this week?
2. Region/dialect: Different parts of English-speaking world use different wording for politeness.
3. Relationship: Is the addressee a peer or a superior?

As a Canadian English speaker, I would say either of your sentences can be used politely, but a few tweaks are needed:

1. "in this week": This is unnatural. Use "this week" or "during the week": both are equivalent.
2. Question form: Typically, a question form might be seen as less pushy. (Tied to factors 2 and 3 above.)
3. Replace "when" with "if": This can only be done if you are willing to give the other person an option to not meet this week. (Tied to factors 1 and 3 above.)
4. "your convenient time": This is unnatural. You would use "time convenient for you" if you want to stress on the addressee's convenience or "a convenient time" if you want to talk about a time convenient to both of you.

Putting all this together, one can arrive at many possible sentences. Here are some examples.

I would use this only with someone over whom I have authority:

Let me know when you are free during the week to discuss.

I would use this for a peer or superior if I wanted to give them the option to postpone:

Could you please let me know if you have time to discuss this week?

This is a clever construction that forces the other person to commit to a time this week, but you still make it sound like you are leaving it up to her/him. I would pick one of these if I wanted to push someone with whom I had a formal relationship:

Let me know what time is convenient for a discussion this week.

Could you please let me know what time is convenient this week for a discussion?

Many other answers are possible. We would really need better context to narrow it down.

Answer 2

I would say "Let me know when you are free to discuss (this matter)" is appropriate. When you add in "in this week" or any other time table, it can be viewed as you not respecting their time.