The answer is no. Some arguments against the hypothesis of "string theory as the boundary of $M$-theory"
1) The boundary of an Anti de-Sitter space is a sphere. No consistent quantum theory of strings already exist over spheres.
2) The AdS/CFT duality is gravity/gauge duality. Both string and $M$-theory contain gravity. No single example of gravity/gravity duality has been constructed, so far.
3) The strong coupling limit of the type $IIA$ string is $M$-theory. This affirmation is very precise, and the direction that emerges (the one parametrizing the value of the $IIA$ dilaton) is a circle. I cannot see how does anyone can construct an AdS space as a circle fibration over its boundary.
4) The strong coupling limit of the $E_{8}\times E_{8}$ string is heterotic $M$-theory. The problem here is, again, that it's pretty difficult (if not impossible) to generalize the Horava-Witten construction (two parallel $M9$-planes separated by a distance proportional to the VeV of the heterotic string dilaton) in a hyperbolic space.
5) All the known examples of the AdS/CFT type are constructed as the near horizon geometry of a stack of $D$-branes. You are suggesting an $AdS_{11}$/String theory correspondence; well, the problem is that Nham's theorem rule out the possibility of an holographic dual for $AdS_{11}$ because no interacting SCFTs theories exist above six dimensions.
6) Recall that the AdS/CFT correspondence is indeed an AdS $\times X$ /CFT correspondence, where $X$ is some manifold (typically a sphere). Then you are suggesting the existence of a higher dimensional supersymmetric theory. And it is a well known fact that no supersymmetric supergravities exist above twelve dimensions.
7) There are no solutions for the eleven dimensional supergravity in $AdS_{11}$.
8) An overall $RR$ flux is needed for the correspondence to work. What is the analogue in this case? there is none. All the possible central extensions of the eleven dimensional super-Poincarè algebra have been studied, and no one produce a negative value for the cosmological constant.
9) Large $N$-limits are required for the correspondence. What is the propossed limit in this case?
Many more arguments can be elaborated. But I find those enough to ensure the implausibility of the proposal.