Should I tell my recruiter about my interviews at other company and to speed up the process?

So, I cleared one FAANG interview last week and 4 rounds are left from this which the recuiter said he will schedule in upcoming weeks.

I have also cleared one round of another very good MNC. Here I have two round left.

Should I tell my FAANG recruiter about this new company? if yes then what should I say ( what words not to use?).

I don't think I will be clear any of the two , but still if possible I would like to have both offers in my hand at once so that I can decide or negotiate if possible.

Thanks in advance for help , this is my first time job switch thats why dumb question!