TEL AVIV—Israeli tanks and infantry pushed into the outskirts of Gaza City on Monday and severed one of the main roads connecting the northern part of the Gaza Strip to the south, witnesses said, in a major advance that appeared aimed at encircling the enclave’s biggest population center.

A spokesman for the Israeli military declined to discuss the locations of troops, but said operations inside Gaza were expanding as more armored vehicles, artillery and combat engineers joined the fight aimed at dismantling the Palestinian Islamist militant group Hamas and evicting it from Gaza.