Stuart Lancaster wants Farrell to become a permanent member of the England management team, following his successful secondment during the RBS 6 Nations.
The Rugby Football Union have opened negotiations with Saracens over Farrell, who still has two years remaining on his contract with the Aviva Premiership champions.
Talks will resume between the two parties on Monday and Griffiths does not want the situation to drag on into next week, with Saracens facing Clermont Auvergne in the Heineken Cup quarter-final next Sunday.
"We would like to (resolve it) because we don't want any 'will he? won't he?' distractions going through the Clermont week," Griffiths said after Saracens' 24-19 defeat to Harlequins at Wembley on Saturday.
"It is like any kind of discussions. You would like an early settlement but we will see. We will have more discussions this week.
"It is not very complicated to work out what the discussions are about but the default position is that he is the head coach of Saracens. That is the default position if the discussions are not successful."
Griffiths does not foresee a situation where Farrell combines his roles with England and Saracens, as Shaun Edwards did with Wales and Wasps.
"We have previously ruled that out because we think it is difficult to do both jobs. Maybe there was a different set of circumstances with Shaun Edwards."
Harlequins opened a 24-12 lead with tries from Jordan Turner-Hall, George Lowe and Danny Care but required an heroic defensive performance to secure the victory.
Lowe produced two brilliant try-saving tackles in the space of three minutes as Saracens threw the kitchen sink at the Premiership leaders.
Eventually, when Harlequins had Joe Marler and Care in the sin-bin, Richard Wigglesworth stole over for the try on his comeback from injury.
Harlequins then saw Nick Easter replace Marler in the sin-bin and they finished the match with 13 men but hung for the victory that opens a six-point gap on top of the Premiership table.
Lowe is already on the radar of England head coach Stuart Lancaster and Harlequins director of rugby Conor O'Shea believes he could make the summer tour to South Africa.
England will be taking around 40 players as they have two midweek fixtures as well as three Tests against the Springboks.
O'Shea said: "I have to pinch myself to think that he is 22.
"George is an outstanding outside centre and hopefully he can put himself on the map (for England) during the end of the season having been unfortunate with injury earlier.
"If the team is being successful, then people get recognised."
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