Middlesex 108 for 3 trail Yorkshire 317 for 7 (Leaning 85, Hodd 51) by 209 runs
Yorkshire and Middlesex will almost certainly have to settle for a Specsavers County Championship draw following rain on day one and more rain and bad light from early afternoon onwards at Headingley on day three.
Both sides will still find themselves in relegation danger in Division One with three matches left to play despite encouraging performances.
Yorkshire travel to the Oval to face Surrey next week and Middlesex host Hampshire at Uxbridge.
Their nerves will be heightened if second-bottom Somerset beat Warwickshire at Edgbaston.
Yorkshire were bowled out for 358 during the first hour of play having started on 317 for 7 before Middlesex responded with 108 for 3 from 32 overs.
There was no play possible beyond 4pm when bad light and then rain came in for the second time in the day, with 56 overs lost to take the match tally to 142.
Having posted four batting bonus points for passing 350, Yorkshire were going to be the most likely winners in this shortened game.
But Middlesex openers Sam Robson and Nick Compton thwarted such ideas with a composed opening partnership of 73 inside 22 overs either side of lunch.
Although overhead conditions have been overcast for large parts of the last two days, the pitch has been as good for batting as at any stage this season in Championship cricket at Headingley.
There was the odd stifled appeal and a couple of extremely well judged leaves that left hands on heads as Robson and Compton blunted the new ball.
The breakthrough came when the excellent Ryan Sidebottom, in his penultimate match at Headingley, rocked back Compton's off stump for 42 with one that held its line from over the wicket as the batsman played down the wrong line.
The veteran left-armer then had Stevie Eskinazi caught well down low at third slip by a tumbling Jack Leaning for five as the score fell to 83 for two in the 24th over.
Only five more balls were possible before the weather intervened for the first time just before 2pm.
When play resumed 90 minutes later, Sidebottom had Robson caught behind for 47 by a tumbling Andrew Hodd as the score fell to 104 for three in the 32nd.
Only two more balls were bowled, with Sidebottom finishing with three for 49 from 14.
The day started with Tim Murtagh taking two of the three Yorkshire wickets to fall, including Liam Plunkett lbw for 34 with the third ball of the day.
Murtagh then had Ben Coad caught at first slip by Adam Voges as the score fell 336 for nine in the 103rd over, the fifth of the morning, as the visitors claimed their third bowling point.
Sidebottom hit two eye-catching boundaries through the covers and to wide long-on to help secure a fourth batting point before James Harris wrapped up the innings when he had Tim Bresnan caught at deep point for 44 in the 112th over.
Murtagh finished with four for 77 from 29 overs, the pick of the Middlesex bowling figures.