The Met Office said an average March-May temperature of 8.97C (48.15F) would be beaten only by 2007 and 2011 in the records, which date back to 1910.
And depending on temperatures in the final three days of May, Scotland could be set for its warmest spring on record.
It would also mean six months of above-average temperatures across the UK.
The figure of 8.97C is based on temperatures up to 28 May and assumes average conditions until the end of the month, the Met Office said.
This would make this year's spring the third hottest in England, Wales and Northern Ireland, while Scotland's current average of 7.63C (45.73F) is narrowly above the current record of 7.61C (44.70F) set in 2011.
UK temperatures from December to April have all been at least one degree Celsius higher than the long-term average, the Met Office said.
Despite its temperature, May has been duller and wetter than average, with just 76% of the sunshine usually expected in the UK as a whole, and 97.7mm of rainfall - 140% of the long-term average.
This month has brought a wide range of weather conditions, from warm sunshine to torrential rain and even reports of a tornado.
After a week of widespread showers, the weekend outlook is brighter for many areas, with forecasters saying it would feel pleasantly warm with light winds in some parts.
The UK is on course for the third warmest spring since records began, forecasters say.