(CNN) - A new week brings the same big problem for South Carolina Gov. Mark Sanford.
Sanford's trying to hold onto his job following his admittance last week of an extramarital affair with a woman from Argentina. The governor also acknowledged at a news conference on Wednesday, upon his return from Argentina, that he did not tell his staff he was out of the country during a five-day period when his location was not known.
Sanford is scheduled on Monday to chair a state budget board meeting, his second public meeting since his return. Sanford apologized to state officials on Friday at a cabinet meeting that was covered live by the national cable news networks, including CNN.
Asked about resigning during a meeting with reporters later on Friday, Sanford said, "You take everything a day at a time. I wouldn't say anything definitive at this point. I say my hunch at a variety of different levels would be to continue on."
The South Carolina governor told the Associated Press on Sunday that he thought about stepping down until close spiritual and political associates urged him to stay on finish out the remaining year and a half of his second and final term as governor. Sanford, who is serving his second term as the state's governor, cannot run for re-election in 2010 because of limits on terms.
"Resigning would be the easiest thing to do," Sanford said in the AP interview, which CNN confirmed with governor's office.
This week several Republican state lawmakers, who are Sanford opponents, could publicly call for the governor's resignation.
Since Sanford's tearful admission last week, two state newspapers have called for him to step down and two state lawmakers have demanded investigations into whether Sanford was derelict of duty by misleading his staff to his location and whether he misused state property or funds to pursue his extramarital affair.
If Sanford were to resign, Lt. Governor Andre Bauer would become governor. Bauer is a fellow Republican but not an ally of Sanford. Bauer is among a number of candidates hoping to succeed Sanford in next year's election.
The woman with whom Sanford had an affair broke her silence Sunday long enough to say she wouldn't discuss her relationship with the now-embattled
In a statement released to Argentina's C5N television network, Maria Belen Chapur said her private life "has already been made too public during these last days, bringing to me even more pain."
Jenny Sanford said Thursday that her husband's political career is "not a concern of mine" and that she'd be just fine - whether or not their marriage survived.
She would not speculate whether her husband would resign as South Carolina governor, saying "he's going to have to worry about that. I'm worried about my family and the character of my children."
–From CNN's Paul Steinhauser and Peter Hamby