Is there a way that I can estimate what the results of a t-test would be in that case?

If you know the covariance (or the correlation) between (Treatment,Control) pairs, then yes, because if we let $T_i$ and $C_i$ be the treatment and control values for the $i$-th subject, then

$\text{Var}(T_i-C_i)= \text{Var}(T_i)+\text{Var}(C_i)-2\text{Cov}(T_i,C_i)$

Since $\text{Var}(T_i)$ and $\text{Var}(C_i)$ are known that leaves only the covariance being needed (and if you know the correlation, the covariance can be computed from the correlation and the two known variances).

If you have some reasonable estimate of the correlation (or bounds on it), you can produce a similarly reasonable estimate of the denominator of the t-statistic (or perhaps bounds on it).

Is there a way that I can estimate what the results of a t-test would be in that case?

If you know the covariance (or the correlation) between (Treatment,Control) pairs, then yes, because if we let $T_i$ and $C_i$ be the treatment and control values for the $i$-th subject, then

$\text{Var}(T_i-C_i)= \text{Var}(T_i)+\text{Var}(C_i)-2\ \text{Cov}(T_i,C_i)$

Since $\text{Var}(T_i)$ and $\text{Var}(C_i)$ are known that leaves only the covariance being needed (and if you know the correlation, the covariance can be computed from the correlation and the two known variances).

If you have some reasonable estimate of the correlation (or bounds on it), you can produce a similarly reasonable estimate of the denominator of the t-statistic (or perhaps bounds on it).