Let $D_{\mathrm{KL}}(P\|Q)$ denote the Kullback–Leibler divergence between discrete probability distributions $P$ and $Q$. It is well-known that the following relation holds between the KL-divergence and mutual information: $$I(X;Y)=D_{\mathrm{KL}}(P(X,Y)\|P(X)P(Y)) \enspace,$$ where $P(Z)$ denotes the probability distribution corresponding to the random variable $Z$.

Now, consider the definition of total variation distance between discrete probability distributions $P$ and $Q$: $$\Delta = \frac 1 2 \sum_x \left| P(x) - Q(x) \right|\enspace.$$

Pinsker's inequality gives the relation between the KL divergence and the total variation distance: $$\Delta(P,Q) \le \sqrt{\frac{\ln 2}{2} D_{\mathrm{KL}}(P,Q)(P\|Q)} \enspace.$$ (The term $\ln 2$ appears since I'm measuring the entropy in bits, while the respective Wikipedia formula uses nats.)

Finally, we note that for any $x,y$, we have ($\delta$ and $\epsilon$ are defined in the question): $$|\delta| = \Big|\Pr[X=x,Y=y]-\Pr[X=x]\cdot\Pr[Y=y] \Big| \le 2\Delta(P(X,Y),P(X)P(Y)) \le 2\sqrt{\frac{\ln2}{2}D_{\mathrm{KL}}(P(X,Y),P(X)P(Y))}=2\sqrt{\frac{\ln2}{2}I(X;Y)}=\sqrt{2\epsilon\ln2} \enspace.$$

PS: This specially appears to be consistent with an example by @whuber (see comments below the question).

Let $D_{\mathrm{KL}}(P\|Q)$ denote the Kullback–Leibler divergence between discrete probability distributions $P$ and $Q$. It is well-known that the following relation holds between the KL-divergence and mutual information: $$I(X;Y)=D_{\mathrm{KL}}(P(X,Y)\|P(X)P(Y)) \enspace,$$ where $P(Z)$ denotes the probability distribution corresponding to the random variable $Z$.

Now, consider the definition of total variation distance between discrete probability distributions $P$ and $Q$: $$\Delta = \frac 1 2 \sum_x \left| P(x) - Q(x) \right|\enspace.$$

Pinsker's inequality gives the relation between the KL divergence and the total variation distance: $$\Delta(P,Q) \le \sqrt{\frac{\ln 2}{2} D_{\mathrm{KL}}(P\|Q)} \enspace.$$ (The term $\ln 2$ appears since I'm measuring the entropy in bits, while the respective Wikipedia formula uses nats.)

Finally, we note that for any $x,y$, we have ($\delta$ and $\epsilon$ are defined in the question): $$|\delta| = \Big|\Pr[X=x,Y=y]-\Pr[X=x]\cdot\Pr[Y=y] \Big| \le 2\Delta(P(X,Y),P(X)P(Y)) \le 2\sqrt{\frac{\ln2}{2}D_{\mathrm{KL}}(P(X,Y),P(X)P(Y))}=2\sqrt{\frac{\ln2}{2}I(X;Y)}=\sqrt{2\epsilon\ln2} \enspace.$$

PS: This specially appears to be consistent with an example by @whuber (see comments below the question).