With Jenson Button and Sergio Perez still well off the pace in qualifying, despite a vain attempt to rush new parts to the racetrack overnight, Whitmarsh was asked whether he had considered stepping down and whether his position had been discussed internally.
"No, I don't believe it's been considered at board level at the moment, as far as I know," he replied.
"I believe in the team, I believe we are going to power through this. So no, I am not considering anything other than getting this team back to where it belongs.
"I've been around in the sport for a long time. I think I've sat here in difficult, dark moments and sat here in good moments as well... I am sure we are going to win some more races and we're going to work hard to do that this year."
Button, the 2009 world champion, won the last race of 2012 in Brazil but has finished no higher than fifth so far after four races this year. Perez's best performance was sixth in Bahrain last month.
McLaren won seven races last season, as many as champions Red Bull, but lost 2008 champion Lewis Hamilton to Mercedes at the end of the year with long-serving technical director Paddy Lowe also departing.
The Woking-based team, 50 percent owned by Bahrain's sovereign wealth fund Mumtalakat, have won more races and drivers' championships than any team other than Ferrari but have not claimed a constructors' crown since their eighth in 1998.
McLaren have periodically gone from boom to bust over the years, winning one race in 1983 and then 12 in 1984, three in 1987 and 15 in 1988, one in 2004 and 10 in 2005.
The team had hoped last month that an upgrade package for Barcelona would narrow the gap at the front but noticeably played down their expectations in the run-up to the race.
Friday practice, wet in the morning, was the equivalent of a bucket of cold water in the face and the team resorted to the seemingly desperate measure of expediting two new front wings from Britain to Barcelona.
They arrived in time for final practice but were not used, and will not be on the cars for the race either, with McLaren saying they could not be sure they would pass scrutineering by the governing FIA.
Asked how it was possible that a team like McLaren could go to all that effort without knowing whether the wings were even legal, Whitmarsh said the FIA equipment was not available to test them and the risk of failure was too great.
"In fairness to the FIA, they have got no obligation to make it available to us early on a Saturday morning," he added. "But that was the chance that we took.
"The tolerancing was sufficiently close that we took a view that this was something we would have to check... without that check, it wasn't prudent to go forward with those wings."
Button, who failed to reach the third and final phase of qualifying on Saturday for the first time this year and will start 14th, put a positive gloss on the situation and cast his mind back to tough times at his previous Honda team.
"We haven't made the step forward everyone expects and we would hope for, but there is so much hard work going on at the factory, I can tell you that," he said.
"It's a very competitive sport and we're not suddenly going to jump to the front of the grid. If we did then the rest of them are doing something wrong at this point in the season.
"The important thing is to keep everyone positive, and I don't think we need to do that because there is a great atmosphere within the team, and I can see us moving forward."