The Argentine star, nicknamed 'The Jewel', joined the Hammers on a season-long loan deal from UAE side Al-Jazira during the summer, and arrived with a strong reputation having starred with River Plate previously.
The 22-year-old has now established himself as a firm fan favourite at Upton Park, and one of the Premier League's newest sensations with a string of fine performances.
With two goals, and assist and a consistent threat in the final third, Lanzini has become a key part of Slaven Bilic's side early in to the season - and are now seeking a permanent deal, according to reports.
The Daily Mirror report that the West Ham hierarchy are concerned that they could lose Lanzini if they wait until the end of his loan spell, and manager Bilic is keen to push through a permanent deal in January to rule out that possibility.
However, the report also states that Al-Jazira want a fee of £8m for Lanzini, which West Ham are hoping to bring down as they consider the current valuation to be too high.
Given Lanzini's displays so far, it's no surprise to see West Ham strongly pursuing a permanent move, but whether they can agree a price with Al-Jazira remains to be seen, with interest in his signature no doubt building ahead of the January transfer window.