Levein is under pressure from many of fans following a poor start to the Ladbrokes Premiership campaign with the Jambos bottom of the table.
Saturday's 3-2 defeat by Motherwell saw protests take place outside the main stand after full time with calls for Levein to be sacked.
However, Budge is standing by the 54-year-old.
"Our Board remains behind the manager, his coaching staff and his team," said Budge, "not because he is “untouchable”, not because he is “all-powerful”, and not because we are “weak”, as has been suggested. We remain behind him because we believe in the plan we have jointly put together.
"We will continue to scrutinise our on-field performances and should we feel our plans are going astray and our approach needs to be adjusted, we will respond accordingly."
In a wide-ranging statement addressing the club's recent form, fan unrest, the club's ambition and media speculation, Budge set out her and the club's stance.
"Like every other Hearts supporter at the moment, I am hugely frustrated with the results, or indeed lack of them, from the first five Premiership fixtures.
"There is no way any of us set out at the start of the season believing we would be languishing in the bottom half of the table at this point, let alone in bottom position.
"Clearly, what we wanted to do, was to have the same kind of start as we had last season, without the catalogue of injuries that then befell us. However, as we all know, that has not happened. So what do we do about it?
"What we do not do is over-react, throw all of our plans out of the window and start again!
"I am frequently asked in what way running a football club is different from running a “normal” business. While there are many differences, I firmly believe that one thing is the same, no matter what business you are running. In simple terms, it is essential for the board to have a plan and to continuously measure performance against that plan, year on year, learning from past mistakes and adjusting our plans accordingly. That is precisely what I, and the Board, have been doing. "
"In season 17/18, when we finished 6th what did we do? We discussed the situation at length and we responded by investing heavily during the transfer window, to re-shape the team. In season 18/19, we, like every other Hearts supporter were delighted that our actions seemed to be paying dividends when we were riding high in the early part of the season. Then disaster struck and we suffered an unprecedented level of injuries. This meant that for the second year in succession, it looked, as though, achieving our football target of finishing in the Top 4 was at risk.
"What did we do? We went back to the planning board. In January, 2019, well before the end of the season, we took action and modified our plan to focus our investment entirely on strengthening our football operation. We put a halt to all new infra-structure developments, ring-fencing our resources to secure those players we believed were key to our future success, while recognising that we would, once again, aim to further strengthen and expand our first team squad in the summer. This we did.
"We all know that while some of these new signings have been settling in and while we are still carrying injuries to a number of our key players, our performances in the early part of the season have been hugely disappointing. However, the Board firmly believes it is too early to be panicking and tearing up our plans."
Budge outlined in the statement three key reasons why she saw no need to dismiss the manager at this moment in time.
"We have a very strong squad, arguably the strongest we have put together over the last 6 seasons," she insisted.
"We have the mix of youth, experience and quality that we have been striving for and building towards since we returned to the Premiership in 2015.
"We have still not seen our first choice starting XI playing this season, as we are still awaiting the return of a number of key players.
"Of course, we hear the criticism: you have wasted money, you have got too many things wrong, you have made too many bad decisions, you have given Craig Levein too long, etc, etc.
"Have we made mistakes at times? Unquestionably, yes. However, running a football club, like any business, is not an exact science. None of us will get every decision right, and it is so easy to be wise after the event and say "I would never have done that!"
"The simple fact is that we plan, make decisions and review on a continuous basis, focusing on how/what we can do better to get to where we want to be.
"In short, we have invested heavily over recent years, never more so than this year, to build what we believe will be a winning squad and we remain confident that this investment will pay dividends. We have planned in a specific way for this season, including catering for injuries, etc., as they come along. My message is simple. I would urge you to continue to support the team while we work through this difficult period.
"Be assured that we are not ‘burying our heads in the sand’; we are in constant dialogue about performances and what is going wrong on the pitch; and we are waiting (im)patiently for the return from injury of some of our key players, and no-one is more impatient than those players themselves."