I'll do my best to answer broadly, but I'd highly suggest better defining your use of the term "serious attempts" for a more precise response.
No proposed amendments concerning impeachment have made it to the ratification process
Proposed Constitutional Amendments
According to the National Archives data set Amending America: Proposed Amendments to the United States Constitution, 1788 to 2014 there have been 17 proposed amendments with the word impeach in their titles or descriptions ranging in date from 1788 to 1989. A quick review narrows the list to 7 that appear to have some proposed impact to the Presidency.
The titles of those 7 are:
- Courts for trial of impeachments
- Trial of impeachment
- Trial of impeachments by some tribunal other than the Senate
- Term of office: Removal: Impeachment
- Providing for trial and removal of civil
officers of United States by means other than impeachment
- Impeachment, permit conviction upon concurrence of a majority of the
Congressional Approval and State Ratification
None of these made it through to the states to be ratified