monica,
The standard version is "I am terribly sorry."
" I am terrible sorry" is used conversationally by certain segments of the population in the U.S.
Other variants:
I am dreadful sorry.
I am awful sorry.
I am frightful sorry.
'Terrible' is considered a quasi-adverb, i.e., it can be used as an adverb while properly speaking it is not.
"Terrible".....to an extreme degree, terribly.
Webster's 3rd Int.
"Terrible"......terribly
Oxford 2nd ed.