You may want to check it the 1Password. Although not free, it provides 2FA integration on both browser and phones.

On the other hand... it may not be the brightest idea to have both password manager AND 2FA in the same baseket, but it may be a good compromise for a lesser secure-demanding services (i.e. those that, if breached, won't affect you in any way other than internet points; think forums, reddit and such).