Hearts centre-back Webster was an unused substitute in the losses to Wales and Belgium which have put pressure on Levein, whose Scotland side sit bottom of World Cup qualifying Group A.
"Like anything you need continuity in any walk of life, club level or international level. You need stability," said Webster, who first signed for Hearts from Arbroath by Levein.
"Nobody wants to be in this situation and it's going to take a lot of hard work and effort to get out of it."
Webster maintains the Scotland squad is a talented one, with proven performers in the Premier League and at the top of the Championship, but is unable to explain the poor run of results.
"The quality in the squad now is as good as it has been in the recent past," he said.
"The level the players are playing at has definitely improved.
"If you could put your finger on it, you would, and you'd rectify what's gone on.
"It's difficult at international level to score the goals you need.
"During the game you might get two or three opportunities and you need to take them."
Asked whether the players should take some responsibility for Scotland's predicament, Webster referred to the home draws with Serbia and Macedonia and rued Steven Fletcher's disallowed goal against Wales, who won in Cardiff last Friday.
"The two home games, on a different day, we probably could and should have won both games," Webster added.
"We had opportunities to score goals in both of them. Going down to Wales on the Friday, the squad as a whole were very optimistic about winning that match.
"From a results point of view, disappointing, especially on the Friday night. People have seen the replays of the goal that never was and if you get 2-0 up the game is finished in my opinion.
"We knew going to Belgium was going to be extremely difficult, but the boys gave a good account of themselves."