The International Atomic Energy Agency (IAEA) has for months been asking to visit the site, warning of "the very real risk of a nuclear disaster".
On Monday Grossi said "the day has come" and that an IAEA support and assistance mission is "now on its way".
On Twitter the IAEA director general said the team from the UN nuclear watchdog would arrive at the power plant "later this week".
The United Nations has called for an end to all military activity in the area surrounding the complex.
Ukraine initially feared an IAEA visit would legitimize the Russian occupation of the site before finally supporting the idea of a mission.
The G7 industrial powers on Monday demanded access "without impediment" for the IAEA team.
They must be allowed to "engage directly, and without interference, with the Ukrainian personnel responsible for operating these facilities", the G7 Non-Proliferation Directors Group said in a statement.
But Ukraine's foreign minister, Dmytro Kuleba, said in Stockholm: "This mission will be the hardest in the history of the IAEA, given the active combat activities undertaken by the Russian federation on the ground and also the very blatant way that Russia is trying to legitimize its presence".
Last week the advisor to the Ukrainian energy minister said she was skeptical the team would even reach the plant.
Advisor Lana Zerkal told Ukraine's Radio NV that Russia was "artificially creating all the conditions so that the mission will not reach the site", despite formally agreeing to the inspection.
Ukraine was the site of the world's worst nuclear catastrophe in 1986, when a reactor at the northern Chernobyl plant exploded and spewed radiation into the atmosphere.
Experts say any leak at Zaporizhzhia would more likely be on the scale of the 2011 Fukushima disaster in Japan.
Energoatom warned on Monday that any leak would scatter radiation over swathes of southern Ukraine and south-western regions of Russia.
Kyiv suspects Moscow intends to divert power from the Zaporizhzhia plant to the Crimean Peninsula, annexed by Russia in 2014.
But Russia insists Ukraine is responsible for shelling around the complex.
Russia's defense ministry said on Saturday that Ukrainian forces had "shelled the territory of the station three times" from the town of Marganets across the Dnipro River.
The ministry accused Kyiv of "nuclear terrorism" and said shells had landed near areas storing fresh nuclear fuel and radioactive waste.
Radiation levels at the plant "remain normal", it said.
But residents in the Ukraine-held areas around the plant are being equipped with iodine pills to reduce the medical risk of radiation in the event of a disaster.