Probably you are looking for 
PUT /rest/api/1.0/projects/{projectKey}/repos/{repositorySlug}/browse/{path:.*}
From the documentation of API
PUT
  /rest/api/1.0/projects/{projectKey}/repos/{repositorySlug}/browse/{path:.*}
Update the content of path, on the given repository and branch. This
  resource accepts PUT multipart form data, containing the file in a
  form-field named content. An example curl request to update
  'README.md' would be: 
curl -X PUT -u username:password -F  content=@README.md  -F
    'message=Updated using file-edit REST API'  -F branch=master -F 
    sourceCommitId=5636641a50b http://example.com/rest/api/latest/projects/PROJECT_1/repos/repo_1/browse/README.md
- branch: the branch on which the path should be modified or created
- content: the full content of the file at path message: the message
  associated with this change, to be used as the commit message. Or null
  if the default message should be used.
- sourceCommitId: the commit ID
  of the file before it was edited, used to identify if content has
  changed. Or null if this is a new file.
The file can be updated or
  created on a new branch. In this case, the sourceBranch parameter
  should be provided to identify the starting point for the new branch
  and the branch parameter identifies the branch to create the new
  commit on.