Yes, it is even possible without interaction (nothing Bob needs to send to Alice). The method is called "ring signature".
Let's say she wants to sign a message like "I am Alice an hereby proof to Bob that I know one of the keys".
She hashes it to get $m$.
Alice now generates a random value $r_i$ for every public key $k_i$ and encrypts them to get $y_i$.
Note that they all $y_i$ are unpredictable pseudorandom values. The only $y_i$ she can choose is the one that belongs to her key $k_j$, she just chooses $y_j$ and sign it to get $r_j$ ($r_j$ looks like every other random data)
Now she can choose $y_j$ so that the xor of all $y_i$ equals $m$.
She sends the message and all the $r_i$ to Bob (if the order is not clear, add a note which $r_i$ belongs to which key)
To verify, Bob just encrypts every $r_i$ with the public key $k_i$ to get the $y_i$, xors them all and checks if it equals $m$.
Since all $y_i$ are like random numbers, when you don't know the key, there is no way to fake a signature without knowing a private key.
Additionally there is no way to tell which $y_i$ and $r_i$ was not randomly generated, because they all look random.
I forgot the symmetric encryption step in the ring signature. Between the xor steps symmetric encryption should be applied. This still allows allice to recover the $y_i$ she needs, but makes attacks harder.
For more details look at Wikipedia