If it stays goalless at full-time, then it will see Brendan Rodgers's side through 1-0 on aggregate following their victory in the first leg in Scotland last week.
The first chance of the half at Anfield fell to Hearts as early as the fifth minute, when Callum Paterson tried a half-volley from outside the Liverpool penalty area, but he failed to find the net.
However, the hosts began to dominate proceedings thereafter and could have drawn first blood three minutes later through Adam Morgan.
Four minutes later Morgan found the target, but the assistant referee ruled out his goal as the ball had gone out when Suarez delivered the cross.
Much against the run of play, Hearts could have taken the lead moments later when David Templeton carved out a brilliant cross from the right, but John Sutton failed to keep his close-range header on target.
Templeton was active again in the 39th minute when he tried his luck from outside the Liverpool 18-yard box, but could not keep his shot on target.