Get the rent agreement in your name. But first confirm that with your employer bcz some companies keep a cap on distances of rented property.
Like you say you work in Mumbai but claim HRA for house in Delhi, most probably company might not agree.
In such cases you can go through section 80GG. For 80gg you have to give up HRA. Learn about it, calculate values for both then decide accordingly.
Disclaimer: I am not a tax consultant or a CA. Sawaari saaman ki khud jimmedar hai