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).
With 1Password taking the typically suicidal approach of re-writing everything to fix some abstract problem that doesn’t really exist, I wouldn’t recommend anyone go there.
Non-nerds get very confused with 1Password, as you can easily have multiple versions that work slightly differently at once.
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).