With the pre-orders for the eagerly-awaited Verizon Wireless iPhone 4 kicking off 3 a. m. Eastern on Thursday, February 3, the initial demand for the handset was notably strong; and the carrier’s website reflected a heavy initial demand for the device.

Verizon also reportedly have a rather smooth system on the first day of the online sales of the iPhone; even though the pre-order attempts of some enthusiastic buyers were hindered by error messages.

According to a Verizon spokeswoman, the problems faced by some people were “account-specific”; with a would-be customer probably not allowed to pre-order the iPhone due to certain specific reasons – like, not being the primary holder of a family or business account.

Meanwhile, thanks to the strong initial demand for Verizon’s forthcoming CDMA iPhone 4, both Apple and Verizon have already run out of pre-sale stock of the new handset on the first day itself. Customers are being asked to wait till February 9 for the online purchase of the handset or till February 10 for in-store purchases.

According to an earlier-this-week AppleInsider report, Verizon’s Chief Operating Officer John Stratton had anticipated the demand for the Verizon iPhone will be “unprecedented”; and the carrier had sent out a companywide email urging its employees to refrain from the immediate purchase of the iPhone either for themselves or for their families.