"If the winds do arrive, drivers of high-sided vehicles and motorcyclists are encouraged to avoid the bridge and use the Western Ring on State Highways 16 and 18."
The agency said it would not hesitate to close the bridge on safety grounds if needed.
Meanwhile, Vector workers were gearing up for a busy night.
Vectors crews have been on standby to tend to power outages in Auckland caused by severe winds that could reach 120km/h - while residents have been urged to keep devices charged and a torch handy in case they find themselves in the dark.
Vector head of field services Marko Simunac said crews were ready to attend to any outages if needed.
"We understand that power outages can be frustrating, and so our team will work as quickly and safely as they can to get the power back up and running if this happens.
"If conditions do worsen, we may need to stand our crews down for safety. Please try to be patient if this happens and you are experiencing an outage."
The fastest way to report a power outage was via the "outage centre" on Vector's website, he said.
A "strong wind watch" is also in place from 3am to 4pm on Tuesday for Northland, Great Barrier Island and the Coromandel Peninsula - warning of southwest winds that could approach severe gale in exposed places.
A number of overnight and Tuesday morning road snowfall warnings have been issued - including for the Desert Rd where up to 2cm was expected to settle about the road summit.
Up to 4cm above 800m on Milford Rd, which is closed from Hollyford to Gulliver River Bridge. It was expected to reopen on Tuesday morning.
Up to 12 cm of snow was expected above 700m on Porters Pass above 700m, up to 2cm about the summit of SH6 / Haast Pass, up to 10cm above 800m on SH8/ Lindis Pass above 800m and up to 12cm above 800m on Crown Range Rd.
Road snowfall warnings were also in place for State Highway 7 / Lewis Pass, where up to 4cm could accumulate above 800m.
