Add existing custom field to project screen via REST API

Using Rest API i would like to add an existing custom field to the desired project screen. This is specifically for Jira cloud version.
Use case flow:

  1. I will take an input from user for project selection
  2. drop down option on available custom field in the system
  3. Add the custom field to the project screen.

I am not able to find an REST API to add a custom field to the project screen. If anyone has developed such feature using API please do share the logic or API’s.

Fields are actually not added directly to screens, but to tabs within screens. This is the API to add a field to a tab: Add screen tab field.

Awesome, thank you so much for the guidance, let me explore a lil more and get back to you in case of any questions.