There is workaround for this problem.
- Install Twitter android app in your mobile
- Login to your account
- Under the options menu, select Settings and then select your account
- Now select Security, this will prompt you to add phone number, then click OK.
- You will be redirected to a browser page, enter your mobile number there and save.
- Now you can go back to your account in normal desktop browser and change the access level settings to read and write.
Cheers!