Summary:
While integrating Web TimeSheet with Microsoft Project, it may throw the message This module currently does not have permission to access Web TimeSheet Would you like to authorize this module now? after you type the login credentials of Web TimeSheet in the Microsoft Project Integration Manager. When you click on the ‘OK’ button, you may encounter this error 
An error occurred while attempting to connect to Web TimeSheet. Authentication failure.

Resolution:
Integration Manager for Microsoft Project is most likely not enabled under Integrations tab of Web TimeSheet. Enable it and then type in the login credentials of the Web TimeSheet in the Microsoft Project Integration Manager, it would integrate with Web TimeSheet.