This Letter presents measurements of correlated production of nearby jets in Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}} = 2.76$ TeV using the ATLAS detector at the Large Hadron Collider. The measurement was performed using 0.14 ${\rm nb}^{-1}$ of data recorded in 2011. The production of correlated jet pairs was quantified using the rate, $R_{\Delta R}$, of "neighbouring" jets that accompany "test" jets within a given range of angular distance, $\Delta R$, in the pseudorapidity--azimuthal angle plane. The jets were measured in the ATLAS calorimeter and were reconstructed using the anti-$k_{t}$ algorithm with radius parameters $d = 0.2$, $0.3$, and $0.4$. $R_{\Delta R}$ was measured in different Pb+Pb collision centrality bins, characterized by the total transverse energy measured in the forward calorimeters. A centrality dependence of $R_{\Delta R}$ is observed for all three jet radii with $R_{\Delta R}$ found to be lower in central collisions than in peripheral collisions. The ratios formed by the $R_{\Delta R}$ values in different centrality bins and the values in the 40-80 % centrality bin are presented.