To connect existing bitbucket repository the easiest solution is to use SSH as appeared.
Firstly, generate on your linux machine the ssh keys (public and private), use for this keygen-ssh command.
There is a nice explanation of this on the atlassian website, but it was not clear enough for me what I should do next.
As appeared it is pretty simple – from the main webpage go to the personal settings menu and add your public key from there.
Now just either clone it using ssh, either change the type from https to ssh and it all works, no need to enter password.