The England U-21 ace has emerged as on of the Premier League’s top young talents this season after being thrust into first-team action by Sam Allardyce.
The Hammers chief took a gamble on the once troubled star, whose career looked to be in tatters after Manchester United lost patience with him due to off-field troubles.
Although Morrison has thrived under ‘Big Sam’, he has been linked with a move away from Upton Park in recent weeks, alerting Meulensteen.
The Dutch coach worked with the 20-year-old while at United, and is said to have a good relationship with him.
He is keen to take advantage of this, and is hopeful that he can lure him across London in a bid to help the Cottagers avoid relegation.
Morrison currently earns £15,000-per-week on a contract that expires at the end of next season.
Fulham are confident that a £5m offer may be enough, with West Ham known to be hoping to invest heavily in their squad in the coming weeks and wary that they could lose their starlet for free in 18 months’ time.